Differential Equations

Table of Contents

Stuff

Spectral method

A class of techniques to numerical solve certain differential equations. The idea is to write a solution of the differential equation as a sum of certain "basis functions" (e.g. Fourier Series which is a sum of sinouids) and then choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

Series solutions

Definitions

Suppose we have a second-order differential equation of the form

differential_equations_32afd12e78439f2fe98ae2d25608d54b67a02456.png

where differential_equations_a9090c77ce9916955c745920bf8f134a0932d59d.png, differential_equations_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png and differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png are polynomials.

We then rewrite the equation as

differential_equations_dcab0299f5da0de7fec0058a96f6fe98c01c3d40.png

Then we say the point differential_equations_c462c980a45481116745a8647094b2b1d245df0f.png is

  • an ordinary point if the functions differential_equations_1090411202281a5f6bb8940a3fd2a5cc9a2d2530.png and differential_equations_0cd479d3312321c8a6a3c7577eec10f3ea921496.png are analytic at differential_equations_c462c980a45481116745a8647094b2b1d245df0f.png
  • singular point if the functions aren't analytic

Convergence

Have a look at p. 248 in "Elementary Differential Equations and Boundary Problems". The two pages that follow summarizes a lot of different methods which can be used to determine convergence of a power series.

Systems of ODEs

Overview

The main idea here is to transform some n-th order ODE into a system of differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png 1st order ODEs, which we can the solve using "normal" linear algebra!

Procedure

Consider an arbitrary n-th order ODE

differential_equations_3495b52b84052adaccd67cc003514590f0a76f67.png

  1. Change dependent variables to differential_equations_c953d0cb8837aa596770ced46340e64cf896e661.png :

differential_equations_59725a2d006e877c2cd63902803c5e378f2623c1.png

  1. Take derivatives of the new variables

differential_equations_a26df271e4ba83dbe9c7579d0e702e1a1379baf6.png

Homogenous

Fundamental matrices

Suppose that differential_equations_7d0cc5296234a3fc52e5ceae574d1600b1a0dd3b.png form a fundamental set of solutions for the equation

differential_equations_b0893d6ec3dec18aef36fc152d7b8c34d9e33fdb.png

on some interval differential_equations_d285471c92cf9470e60690b66c0c6734abf08ac9.png. Then the matrix

differential_equations_a73026605c549a8c1033bb086bfd3a891c94d433.png

whose columns are the vectors differential_equations_7d0cc5296234a3fc52e5ceae574d1600b1a0dd3b.png, is said to be a fundamental matrix for the system.

Note that the fundamental matrix is nonsingular / invertible since the solutions are lin. indep.

We reserve the notation differential_equations_37d17dff65e4110dde040ea4a4a88525328b4ccb.png for the fundament matrices of the form

differential_equations_96292260d0c2d7f499548f3ed3cb0e5032766274.png

i.e. it's just a fundamental matrix parametrised in such a way that our initial conditions gives us the identity matrix.

The solution of an IVP of the form:

differential_equations_6351afaf8d0a3e510448f81765dbc4a1da2b8a4d.png

can then be written

differential_equations_057b078ba0d987d8c5e578b89e9e31f7a4989423.png

which can also be written

differential_equations_22cb5235d5d2ea332fba7d2bd50a864df5e7ec76.png

Finally, we note that

differential_equations_033e6620741011c8722088620045dc67e40270c0.png

The exponential of a matrix is given by

differential_equations_03b33a8469bdb5b5ceed788bdc2a4c292b654704.png

and we note that it satisfies differential equations of the form

differential_equations_a0f7c50a72658d188f68f3d04ff94c11787c29be.png

since

differential_equations_0c41016eca2ce8e6c3bf25e2939ea260ab338d43.png

Hence, we can write the solution of the above differential equation as

differential_equations_22cb5235d5d2ea332fba7d2bd50a864df5e7ec76.png

Repeating eigenvalues

Suppose we have a repeating eigenvalue differential_equations_3b77f2d6624396350dc5d7749e418480a37b42f5.png (of algebraic multiplicity 2), with the found solution correspondig to differential_equations_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is

differential_equations_dd06a0ad35918d85793d56bb34ce0a6547d57f90.png

where differential_equations_9761adf768a249d912eaf7d9c7abf2e53d7f5e9f.png satisfies

differential_equations_670ad3a6438415be1936fb76a669e1d954848199.png

We then assume the other solution to be of the form

differential_equations_08367f88f0b708854b08f683c443a661e5366314.png

where differential_equations_7fa6d5539258f1df66040916ec01134f8545cd0f.png is determined by the equation

differential_equations_1804b225f6d92655e3b8afd5c999849a31364b15.png

Multiplying both sides by differential_equations_b31fcfe89d078ecf840ace7dbb24d95c1dcf1849.png we get

differential_equations_f80401d2ea6baebf76f4507d71b64fd27982f0ec.png

since differential_equations_a66115c87ce9ed73306b30fcbf68725fbb404044.png by definition. Solving the above equation for differential_equations_7fa6d5539258f1df66040916ec01134f8545cd0f.png and substituting that solution into the expression for differential_equations_977a89128e7fd58f5fb9b8bf811d297f0e810bb1.png we get the second solution corresponding to the repeated eigenvalue.

We call the vector differential_equations_7fa6d5539258f1df66040916ec01134f8545cd0f.png the generalized eigenvector of differential_equations_5621442eb84314df5b6d4f90193ccefd86434bb0.png.

Non-homogenous

We want to solve the ODE systems of the form

differential_equations_1e4ae95e119ba04d0a210c783318778d85f21d92.png

Which has a general solution of the form

differential_equations_e3213ea2d54295bc973d9eec185bc78183d35e08.png

where differential_equations_546d81e8adb48bbea83cb7f08b8f8108ff9ffd1f.png is the general solution to the corresponding homogenous ODE system.

We discuss the three methods of solving this:

  1. Diagonolisation
  2. Undetermined coefficients
  3. Variation of parameters
Diagonolisation

We assume the corresponding homogenous system is solved with the eigenvalues differential_equations_a77df1896cf85b639df10d71d8757cdcef2bd181.png and eigenvectors differential_equations_903b1a846e2f046894c2f1a522e2ab903cffa162.png, and then introduce the change of variables differential_equations_e494ba02417bcf23fc349b8505c408d658dc29a8.png :

differential_equations_a0d69d731efa9dbd3b25635903f4f5c77cedee56.png

which gives us a system of differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png decoupled systems:

differential_equations_59c20ed7f35f71e769ba332e835932c918f3b61a.png

and finally we transform back to the original variables:

differential_equations_6e94e4334b916817b0551c622bb66ab554c45f08.png

  • Example

    Consider

    differential_equations_e4ad01cbfbc81e56c156be6522e5bfc1e22c2d2d.png

    Since the system is linear and non-homogeneous, the general solution must be of the form

    differential_equations_b30e92a86bafb43890a785e51d8e0193774a4751.png

    where differential_equations_d19d8b24607400b86827bd94e97be31b3a7c7e23.png denotes the particular solution.

    The homogenous solution can be found as follows:

    differential_equations_e6ee083faacbe27b8dd2b2cd438d9938a4845494.png

    When differential_equations_72470909a9a288c2923a76115e45d530b10e0125.png: if differential_equations_f14c1d85b8c2f622614fea2e7ffc0bc1ff35d048.png, hence we choose

    differential_equations_d0dae29a5874b221712c98766613751b9a34b0e9.png

    When differential_equations_2a0fd731feddaa44f2210c4aa8c62656275a8a75.png: if differential_equations_a0b6e2490f698eca0c0423bc81b628420be46f9c.png, hence we choose

    differential_equations_9d22c3e1c7d43d7403369aae7fb71d12bf6d0f6d.png

    Thus, the general homogenous solution is:

    differential_equations_e3ded64c55ac2fa778f0969b2056621a186bf8d3.png

    To find the particular solution, we make the change of variables differential_equations_abd6f0dab49533c20fc1c60f6f96f56eca0e58bc.png:

    differential_equations_ec8a0bb908a82eaafee154433c128272562c8916.png

    With the new variables the ODE looks as follows:

    differential_equations_cd8cfd60ebff3e166ad6d0cbcf9b4a1108953999.png

    Which in terms of differential_equations_1bd94e5124e81dc58df7a7cbc833b80250e8ec3d.png is

    differential_equations_a4448cafffdbf4eb1f72bf9a36a353174cd05d0e.png

    Which, by the use of undetermined coefficients, we can solve as

    differential_equations_dd0d3915fd087b0b954b4feae69d75f75860d62a.png

    Doing "some" algebra, we eventually get

    differential_equations_1edfa5c903de2dbc7bbc34c0eef5af5acb124f4a.png

    Then, finally, substituting back into the equation for differential_equations_e494ba02417bcf23fc349b8505c408d658dc29a8.png, and we would get the final result.

Undetermined coefficients

This method only works if the coefficient matrix differential_equations_4d73c3f5523951e3a5b489033b204ea971c53168.png for some constant matrix differential_equations_5621442eb84314df5b6d4f90193ccefd86434bb0.png, and the components of differential_equations_c013c1ad8a456487c37de126d34248d793fec7ff.png are polynomial, exponential, or sinusoidal functions, or sums or products of these.

Variation of parameters

This is the most general way of doing this.

Suppose we have the n-th order linear differential equation

differential_equations_f103fe99a798216f81d52517845ba029872b604c.png

Suppose we have found the solutions to the corresponding homogenous diff. eqn.

differential_equations_2fcc97f31a974899c651901abe8ecc2e6e247921.png

With the method of variation of parameters we seek a particular solution of the form

differential_equations_cfe0839ef881ded0cb55e9485ab319a526f0d174.png

Since we have differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png functions differential_equations_933c36933a4f59cb3d5100e48049f9f9d406736a.png to determine we have to specify differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png conditions.

One of these conditions is cleary that we need to satisfy the non-homongenous diff. eqn. above. Then the differential_equations_97e10175daca76010e298f70cce0826a16a6c021.png other conditions are chosen to make the computations as simple as possible.

Taking the first partial derivative wrt. differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png of differential_equations_858f50a429026dc5a18e911c0d6cfc2bc4bd18fe.png we get (using the product rule)

differential_equations_ff4e6fad3c249ba027bb1c3cd4bfbb81febdf872.png

We can hardly expect it to be simpler to determine differential_equations_858f50a429026dc5a18e911c0d6cfc2bc4bd18fe.png if we have to solve diff. eqns. of higher order than what we started out with; hence we try to surpress the terms that lead to higher derivatives of differential_equations_f42b50c6619721049cd9d758da6596ddcb761f22.png by imposing the following condition

differential_equations_aed3618552c1381943099efaba238055a06e78c3.png

which we can do since we're just looking for some arbitrary functions differential_equations_e5d7a7637b3d17e9a0a9d07aea011ca76af56ada.png. The expression for differential_equations_98a8bb3ff68141b85cc65aa358def884835dc13a.png then reduces to

differential_equations_a101a393276864856d2af4c8587cdbfe38b6f373.png

Continuing this process for the derivatives differential_equations_abfb326c0db01e6b0326f788cf4430551a4575a8.png we obtain our differential_equations_97e10175daca76010e298f70cce0826a16a6c021.png condtions:

differential_equations_1fbcebd6d54fe3cfb1adcf865e0a21e710961e01.png

giving us the expression for the m-th derivative of differential_equations_858f50a429026dc5a18e911c0d6cfc2bc4bd18fe.png to be

differential_equations_d5d00c3789dac726d871e3420e7649819990759c.png

Finally, imposing that differential_equations_74b70f16fbcd4161d22c21012510a32d689a7c27.png has to satisfy the original non-homogenous diff. eqn. we take the derivative of differential_equations_e0c9f49c215f297068d1df67fcbbf7ab430413ca.png and substitute back into the equation. Doing this, and grouping terms involving each of differential_equations_fe4fbcb8d7952b2fb6ba6a376dda9c1f28dce4f1.png together with their derivatives, most of the terms drop out due to differential_equations_fe4fbcb8d7952b2fb6ba6a376dda9c1f28dce4f1.png being a solution to the homogenous diff. eqn., yielding

differential_equations_79a1c420d9948b45971b08a8bc06f329fee39190.png

Together with the previous differential_equations_97e10175daca76010e298f70cce0826a16a6c021.png conditons we end up with a system of linear equations

differential_equations_431975d724466f78a4f3ce3bc591a0cf11916b3c.png

(note the differential_equations_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png at the end!).

The sufficient condition for the existence of a solution of the system of equations is that the determinant of coefficients is nonzero for each value of differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png. However, this is guaranteed since differential_equations_d6f3123e21d52b46d2454b67926e741948352821.png form a fundamental set of solutions for the homogenous eqn.

In fact, using Cramers rule we can write the solution of the system of equations in the form

differential_equations_c27debccda1c13b07fbdffcce43aa69643002589.png

where differential_equations_654b7fd5d40298e1fb9b8222fd2990a4991965b9.png is the determinant of obtained from differential_equations_f53cf2c3985c80987bd9707fd5a0b28594db886f.png by replacing the m-th column by differential_equations_eefd56391990a2f1856893cbcc51ce16c3e5deac.png. This gives us the particular solution

differential_equations_504cdbd7254db089ed0054bc908b5a6a8a3489d3.png

where differential_equations_0eca193b6f6cd17cb61e1065c28719e1d5237614.png is arbitrary.

Assume that a fundamental matrix differential_equations_975a9e39bc0c78209c2debef91acb95f780019c0.png for the corresponding homogenous system

differential_equations_118f696cd2d78c7cd35adc50b4150c4b31523d5c.png

has been found. We can then use the method of variation of parameters to construct a particular solution, and hence the general solution, of the non-homogenous system.

The general solution to the homogenous system is differential_equations_c1dddc286a4e4985d844da0fd9212497012e1850.png, we seek a solution to the non-homogenous system by replacing the constant vector differential_equations_cd8e20dba90a9009c6135d2413dce5bad16f4099.png by a vector function differential_equations_e7f34557de04d397e90f1fca90c47ab088b382d9.png. Thus we assume that

differential_equations_cbcc8d6b909835c00a59c8ca41fa7dfe468dc550.png

is a solution, where differential_equations_e7f34557de04d397e90f1fca90c47ab088b382d9.png is a vector function to be found.

Upon differentiating differential_equations_40fb973cd997849a029e392c385d32d4b8c40196.png we get

differential_equations_3c836c3344f0558a92b3aaa2857d07030a6fba83.png

Since differential_equations_975a9e39bc0c78209c2debef91acb95f780019c0.png is a fundamental matrix, differential_equations_4aece379f887b980253ce27bedb41057e41f33dc.png; hence the above expression reduces to

differential_equations_0b45a92e9204ae2c89865404d56be414a1556709.png

Since differential_equations_975a9e39bc0c78209c2debef91acb95f780019c0.png is nonsingular (i.e. invertible) on any interval where differential_equations_96cecc2c419d74d4370d9fd4350a562817c36e10.png is continuous. Hence differential_equations_75e1c0b95fa5b0bf2cd77a07852ea877c87a04a9.png exists, and therefore

differential_equations_a84c41e77266438f5bd92a23d94590efad6ecd7f.png

Thus for differential_equations_e7f34557de04d397e90f1fca90c47ab088b382d9.png we can select any vector from the class of vectors which satisfy the previous equation. These vectors are determined only up to an arbitrary additive constant vector; therefore, we denote differential_equations_e7f34557de04d397e90f1fca90c47ab088b382d9.png by

differential_equations_697be58c8b83cdf3aaddb52a46ee024b1c147122.png

where the constant vector differential_equations_cd8e20dba90a9009c6135d2413dce5bad16f4099.png is arbitrary.

Finally, this gives us the general solution for a non-homogenous system

differential_equations_3c9373372323d972e4f15d5d3cd1e3c46fc9977e.png

Dynamical systems

In differential_equations_004097ff73cb85a0f596c8a3b60218ece0e16be1.png, an arbitrary autonomous dynamical system can be written as

differential_equations_3b0e3e35ca9e252d294de52297ebf50834b93b2a.png

for some smooth differential_equations_cfa4a3dff9583d4aac6e570179000c79f257d8a4.png, for a the 2D case:

differential_equations_11b788596970ca74ea383b2c647292698e33b800.png

which in matrix notation we write

differential_equations_26509ac89aa550218b38c190ef35ada52f00d864.png

Notation

  • differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png denotes a critical point
  • differential_equations_2d0ef2011075542d0f0bb1a877c678a2b9b698b6.png denotes the RHS of the autonomous system
  • differential_equations_30d6e1b431b4d814b92688147cc68c6e17e99aee.png denotes a specific solution

Theorems

The critical point differential_equations_bba4a319536160111df17ef26752f627b4fbc853.png of the linear system

differential_equations_a0f7c50a72658d188f68f3d04ff94c11787c29be.png

where we suppose differential_equations_5621442eb84314df5b6d4f90193ccefd86434bb0.png has eigenvalues differential_equations_3a7b09555a74a4bfc31c7fdf71bd5ea32e4b0b83.png, is:

  • asymptotically stable if differential_equations_3a7b09555a74a4bfc31c7fdf71bd5ea32e4b0b83.png are real and negative
  • stable if differential_equations_3a7b09555a74a4bfc31c7fdf71bd5ea32e4b0b83.png are pure imaginary
  • unstable if differential_equations_3a7b09555a74a4bfc31c7fdf71bd5ea32e4b0b83.png are real and either positive, or have positive real part

Stability

The points differential_equations_76f22636a99bf067e1362d57e248466b7ba74450.png are called critical points of the autonomous system

differential_equations_7778fe695fd8caa786a2e4969c0030a2dfc1a122.png

Let differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png be a critical point of the system.

  • differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png is said to be stable if for any differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png,

    differential_equations_4c4060eb3971358bdc4ba0de96410c958847b76b.png

    for all differential_equations_809038084ca90dd90f9bde810a570e42ccd59b74.png. I.e. for some solution differential_equations_c73cd3b2b54e5cf43171325d44788fe6bc949dfb.png we parametrize it such that differential_equations_246ee1f7422b09a0dcc3b1258deff27e8353289e.png for some differential_equations_0809b466cd9de765ec30e362bd8646103cb497eb.png such that we stay close to differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png for all differential_equations_809038084ca90dd90f9bde810a570e42ccd59b74.png, then we say it's stable.

  • differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png is said to be asymptotically stable if it's stable and there exists a differential_equations_6eb570890a77ea97d7acaa7799e47ff6bf0afd1c.png s.t. that if a solution differential_equations_30d6e1b431b4d814b92688147cc68c6e17e99aee.png satisfies

    differential_equations_e8a9c89d443c45df785143081eff50523ef43644.png

    then

    differential_equations_da66a1681fb68d3f1a9729dddb9ad6b006fb18a9.png

    i.e. if we start near differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png, the limiting behaviour converges to differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png. Note that this is stronger than just being stable.

  • differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png which is NOT stable, is of course unstable.

Intuitively what we're saying here is that:

  • for any differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png we can find a trajectory (i.e. solution with a specific initial condition, thus we can find some initial condition ) such that the entire trajectory stays within differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png of the critical points for all differential_equations_1452aaed016726fd389a85854c1b53f56436757e.png.

When we say a critical point is isolated, we mean that there are no other critical points "nearby".

In the case of a system

differential_equations_31ce9a8437b9b6fd9fa60f1c631a480464d85316.png

By solving the equation

differential_equations_f46874dce0fabd82278257f72bdf3534784ef558.png

if the solution differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png is a single vector, rather than a line or plane, the critical point differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png is not isolated.

Suppose that we have the system

differential_equations_31ce9a8437b9b6fd9fa60f1c631a480464d85316.png

and that differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png is a isolated critical point of the system. We also assume that differential_equations_68797480dd68ac055a9264b16a16fb4ab35945b0.png so that differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png is also a isolated critical point of the linear system differential_equations_4fa574874ac1f95fbc4d48c3fe47ad411f1b524a.png. If

differential_equations_ee2a9a24d00d41a0717e8678c4b31f4044743806.png

that is, differential_equations_b719dfdbe518b7a0364609c3b39f1e3705058635.png is small in comparison to differential_equations_15093e5edf8e93dea5edab4b375b7833d19c670b.png near the critical point differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png, we say the system is a locally linear system in the neighborhood of the critical point differential_equations_edb6dfb18166d66846f8b68496b5aad6b2043db9.png.

Linearized system

In the case where we have a locally linear system, we can approximate the system near isolated critical points by instead considering the Jacobian of the system. That is, if we have the dynamical system

differential_equations_7778fe695fd8caa786a2e4969c0030a2dfc1a122.png

with a critical point at differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png, we can use the linear approximation of differential_equations_fe0aeda2eb51952e6329b759383f4bcb448137f3.png near differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png:

differential_equations_54c45b1a3ad054059dd658f7c04f82b64f4105cb.png

where differential_equations_6b6ca44314f20654431f1d1d40653db422f9beea.png denotes the Jacobian of differential_equations_fe0aeda2eb51952e6329b759383f4bcb448137f3.png, which is

differential_equations_e0d204a88e1a8098f2ba939f85bef8a36d6fed91.png

Substituting back into the ODE

differential_equations_4e870980e8e62e82bc7c2f715d73c7ee4a8afeff.png

Letting differential_equations_cf6cab6669cf05119f0df17ebbc823ec617bc736.png, we can rewrite this as

differential_equations_b057e6a3883dec6a2d0bf6f6a6965b6743aa8149.png

which, since differential_equations_b3d95573082ebc9a161c4fa3ad0ef92074371c96.png is constant given differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png, gives us a linear ODE wrt differential_equations_17e9f52549b4b54d9708d6330ea265cc1a8a030c.png, hence the name linearization of the dynamical system.

Hopefully differential_equations_18da9e1a1b0a3598cc1f0196daba0a7c8446527a.png is a simpler expression than what we started out with.

General procedure
  1. Obtain critical points, i.e. differential_equations_90de71d05e39346bf300ec3f82272ca05345f34c.png
  2. If non-linear and locally linear, compute the Jacobian and use this is an linear approximation to the non-linear system, otherwise: do nothing.
  3. Inspect the following to determine the behavior of the system:
    • differential_equations_c310063367e3b9fa67584f496758da08b4ead79a.png and differential_equations_7eaf720a65c63823c9ea76f4e948d2bd723af771.png
    • Compute eigenvalues and eigenvectors to obtain solution of the (locally) linear system
    • Consider asymptotic behavior of the different terms in the general solution, i.e. differential_equations_fe23fc7dbc30e208134c6de6bb7bf938a93cbfe8.png
    • differential_equations_1dc6266994616318b4258f2cb89d17cef5c79c87.png which provides insight into the phase diagram / surface. If non-linear, first do for linear approx., then for non-linear system
Points of interest

The following section is a very short summary of Ch. 9.1 [Boyce, 2013]. Look here for a deeper look into what's going on, AND some nice pictures!

Here we consider 2D systems of 1st order linear homogenous equations with constant coefficients:

differential_equations_32a392e1621395a477412e43a466cf3cd937b922.png

where differential_equations_0182f90b7ecffc0719931398d4771b2513b9cea3.png and differential_equations_22fc188c73a86fbd1e84b41b3f1a57d8979ab387.png. Suppose differential_equations_3a7b09555a74a4bfc31c7fdf71bd5ea32e4b0b83.png are the eigenvalues for the matrix differential_equations_5621442eb84314df5b6d4f90193ccefd86434bb0.png, and thus gives us the expontentials for the solution differential_equations_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png.

We have multiple different cases which we can analyse:

  • differential_equations_c34f553bb555d22a970ec1b913ad228b84f490c2.png:
    • Exponential decay: node or nodal sink
    • Exponential growth. node or nodal source
  • differential_equations_7d6bcdf1cc5d2e9d7f4d88d5806be1a5f3bdcac6.png: saddle point
  • differential_equations_e542db33247ebfc1248e2ea0753d5e26248ca1d6.png:
    • two independent eigenvectors: proper node (sometimes a star point )
    • one independent eigenvector: improper or degenerate node
  • differential_equations_46a4f16531c8b2491d0468c2a2151773cc78969e.png: spiral point
    • spiral sink refers to decaying spiral point
    • spiral source refers to growing spiral point
  • differential_equations_d6287016d450dc5186f2e5c8b4b75a3690bacb17.png: center
Basin of attraction

This denotes the set of all points differential_equations_a9090c77ce9916955c745920bf8f134a0932d59d.png in the xy-plane s.t. the trajectory passing through differential_equations_a9090c77ce9916955c745920bf8f134a0932d59d.png approaches the critical point as differential_equations_c07ae07299c6cdac38a6b328ab7017954d05c681.png.

A trajectory that bounds the basin of attraction is called a separatix.

Limit cycle

Limit cycles are periodic solutions such that at least one other non-closed trajectory asymptotes to them as differential_equations_c07ae07299c6cdac38a6b328ab7017954d05c681.png and / or differential_equations_db0ddeed267dc7052a11d8a7f95af6ecf7dacc33.png.

Let differential_equations_e07e26b3570b39bbe5619c4d5d7bb773c03f4c5b.png and differential_equations_fcc839b7f2682f9f3fe1036d4c44f4687dae1194.png have continuous first partial derivatives in a simply connected domain differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

If differential_equations_145190ef4561c69cbe66175acee0a401b8446c1a.png has the same sign in the entire differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png, then there are no closed trajectories in differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

Consider the system

differential_equations_fb73046859ad0104e290d77c3984d879311b5840.png

Let differential_equations_e07e26b3570b39bbe5619c4d5d7bb773c03f4c5b.png and differential_equations_fcc839b7f2682f9f3fe1036d4c44f4687dae1194.png have continious first partial derivatives in a domain differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

Let differential_equations_08c83e340c641173cc389ce3f705e6ab675cf5e2.png be a bounded subdomain of differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png and let differential_equations_91b0712187b2acf842111139f9a1035bb590a8e8.png. Suppose differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png contains no critical points.

If

differential_equations_bf01995a16545d58cc04ae9e7cc158997e6d7e93.png

then

  • solution is periodic (closed trajectory)
  • OR it spirals towards a closed trajectory

Hence, there exists a closed trajectory.

Where

  • differential_equations_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png denotes a particular trajectory
  • differential_equations_d396c75501c956ba9f25327d16861350d4e2fab7.png denotes the boundary of differential_equations_08c83e340c641173cc389ce3f705e6ab675cf5e2.png
Useful stuff

differential_equations_b976c07012094ce9ef6e1186735253108b712386.png

  • Example

    Say we have the following system:

    differential_equations_9636ba6cb0a1e9bca71cc567f838395b1fbd2868.png

    where differential_equations_687a25a6a0baecadd4bd5a5d11216b8a8120b495.png, has periodic solutions corresponding to the zeroes of differential_equations_5b05c8ea129b3aefda7e125c3c88744a924823e0.png. What is the direction of the motion on the closed trajectories in the phase plane?

    Using the identities above, we can rewrite the system as

    differential_equations_119dd66c81c682bbb4f29c78539a84b6d5a2fa4b.png

    Which tells us that the trajectories are moving in counter-clockwise direction.

Lyapunov's Second Method

Goal is to obtain information about the stability or instability of the system without explicitly obtaining the solutions of the system. This method allows us to do exactly that through the construction of a suitable auxillary function, called the Lyapunov function.

For the 2D-case, we consider such a function of the following form:

differential_equations_84f2c6161aec15c768c979fcdfc12f9cd67a462a.png

where differential_equations_e07e26b3570b39bbe5619c4d5d7bb773c03f4c5b.png and differential_equations_fcc839b7f2682f9f3fe1036d4c44f4687dae1194.png are as given in the autonomous system definition.

We choose the notation above because differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png can be identified as the rate of change of differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png along the trajectory of the system that passes through the point differential_equations_bcaca16349a57fa4db86decededc11f33eed0a02.png. That is, if differential_equations_d49c6e991bd82de31fe568d0c6d72c0fd43cd1c5.png is a solution of the system, then

differential_equations_42e30bcbf847ca05b8e094616c35bd1f9a982f9f.png

Suppose that the autonomous system has an isolated critical point at the origin.

If there exists a continuous and positive definite function differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png which also has continuous first partial derivatives differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png:

  • if differential_equations_63a1b129418aa1012b7c401790d25e3d076b1360.png ( negative definite ) on some domain differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png, then the origin is an asymptotically stable critical point
  • if differential_equations_f327ff7c534991bdc6e273e35966eb0dd2cfc797.png ( negative semidefinite ) then the origin is a stable point

Suppose that the autonomous system has an isolated critical point at the origin.

Let differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png be a function that is continuous and has continuous first partial derivatives.

Suppose that differential_equations_5b2ede0807b3750a51121f083fc7035c09c240ba.png and that in every neighborhood of the origin there is a least one point at which differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is positive (negative). If there exists a domain differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png containing the origin such that the function differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png is positive definite ( negative definite ) on differential_equations_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png, then the origin is an unstable point.

Let the origin be an isolated critical point of the autonomous system. Let the function differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png be continous and have continuous first partial derivatives. If there is a bounded domain differential_equations_9410d695a2a2f66fc5677987b385e539fa14fe66.png containing the origin where differential_equations_07aef2edc173e1d36df065ac090bdbfe85c9d58c.png for some positive differential_equations_1641d18cc980f8db14cdff95d7417a8526eef446.png, differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is positive definite and differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png is negative definite , and then every solution of the system that starts at a point in differential_equations_9410d695a2a2f66fc5677987b385e539fa14fe66.png approaches the origin as differential_equations_c07ae07299c6cdac38a6b328ab7017954d05c681.png.

We're not told how to construct such a function. In the case of a physical system, the actual energy function would work, but in general it's more trial-and-error.

Lyapunov function

Lyapunov functions are scalar functions which can be used to prove stability of an equilibrium of an ODE.

Lyapunov's Second Method (alternative)

Notation
Definitions

Let differential_equations_f4bbd0f1c8f3e6c53e494d6fab0ee16b51a34eac.png be a continuous scalar function.

differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is a Lyapunov-candidate-function if it's /locally positive-definite function, i.e.

differential_equations_d7b3d570a1b17171ed1dd4a3708b7970234958f4.png

with differential_equations_da9cb51849b13210fa778a80b9907f86fe90c379.png bein a neighborhood region around differential_equations_c96e378410444fb64d900cb4812e2895ad211d9c.png.

Further, let differential_equations_2b49ffe70338dad2e664ca4a44c2a9653e468dc8.png be a critical or equilibrium point of the autonomous system

differential_equations_590f1a0028a735a1181332dd524efa5470c2b681.png

And let

differential_equations_c43c2f945d3132e05ddb091f6c74d7042ffcd992.png

Then, we say that the Lyapunov-candidate-function differential_equations_83202b174a58234b0c4164fcf4b60cbed4d4539a.png is a Lyapunov function of the system if and only if

differential_equations_c7bffd9ea3d4c672c720ce9cedfc1a793dc9a080.png

where differential_equations_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png denotes a specific trajectory of the system.

For a given autonomous system, if there exists a Lyapunov function differential_equations_83202b174a58234b0c4164fcf4b60cbed4d4539a.png in some neigborhood differential_equations_da9cb51849b13210fa778a80b9907f86fe90c379.png of some critical / equilibrium point differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png, then the system is stable (in a Lyapunov sense).

Further, if differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png is negative-definite, i.e. we have a strict inequality

differential_equations_0c34c48a1aaebb3d61a51c0aa9baf8b72c97fc61.png

then the system is asymptotically stable about the critical / equilibrium point differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png.

When talking about critical / equilibrium point differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png, when considering a Lyapunov function, we have to make sure that the critical point corresponds to differential_equations_c96e378410444fb64d900cb4812e2895ad211d9c.png . This can easily be achieved by "centering" the system about the point original critical point differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png, i.e. creating some new system with differential_equations_3352a3efd3b15b51e3be18a355a00b8c6180aedf.png, which will then have the corresponding critical point at differential_equations_b1089d133c86cc06962ed68741df867bdc2c2975.png.

Observe that this in no way affects the qualitative analysis of the critical point.

Let differential_equations_f4bbd0f1c8f3e6c53e494d6fab0ee16b51a34eac.png be a continuous scalar function.

differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is a Lyapunov-candidate-function if it's a locally postitive-definite function, i.e.

differential_equations_31e45f481dbb2a044d9174897c4d2f616e95b363.png

with differential_equations_da9cb51849b13210fa778a80b9907f86fe90c379.png being a neighborhood region around differential_equations_bba4a319536160111df17ef26752f627b4fbc853.png.

Let differential_equations_2b49ffe70338dad2e664ca4a44c2a9653e468dc8.png be a critical / equilibrium point of the autonomous system

differential_equations_0cba00d351b65f4fe9a8e4424855472cf23ff74a.png

And let

differential_equations_ecf8efe4df3efc461e47408722cac850c3959952.png

be the time-derivative of the Lyapunov-candidate-function differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png.

Let differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png be a locally positive definite function (a candidate Lyapunov function) and let differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png be its derivative wrt. time along the trajectories of the system.

If differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png is locally negative semi-definite, then differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is called a Lyapunov function of the system.

If there exists a Lyapunov function differential_equations_8de89f0b52aab42fb2ed3fc97c9194db2eb4f2df.png of a system, then differential_equations_b3559cecc7f934b3d651b0434deec214437097ab.png is a stable equilibrium point in the sense of Lyapunov.

If in addition differential_equations_a15d0e8cd81585ca18c43f05cb3e8beba3ad8fd9.png and differential_equations_83e3d3d40de461b6b269ec6270e60b276cea7952.png for some differential_equations_5926a2dcef2ea762e4e3fb75bcf57e6712557db8.png, i.e. differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png is locally negative definite , then differential_equations_2b49ffe70338dad2e664ca4a44c2a9653e468dc8.png is asymptotically stable.

Remember, we can always re-parametrize a system to be centered around a critical point, and come to an equivalent analysis of the system about the critical point, since we're simply "adding" a constant.

Proof

First we want to prove that if differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is a Lyapunov function then differential_equations_2b49ffe70338dad2e664ca4a44c2a9653e468dc8.png is a stable critical point.

Suppose differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png is given. We need to find differential_equations_0809b466cd9de765ec30e362bd8646103cb497eb.png such that for all differential_equations_e377f17dacc42c3533502099ffd0b418f1b29aa9.png, it follows that differential_equations_c7f6c2d851f14e741aac00d97c6f518dab93cf7e.png.

Then let differential_equations_13edae732701360fe559f5874de0e1460b42172a.png, where differential_equations_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png is the radius of a ball describing the "neighborhood" were we know that differential_equations_83202b174a58234b0c4164fcf4b60cbed4d4539a.png is a Lyapunov candidate function, and define

differential_equations_e709db954be1a2c6759c8db91cf456c0b9ca833f.png

Since differential_equations_83202b174a58234b0c4164fcf4b60cbed4d4539a.png is continuous, the above differential_equations_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is well-defined and positive.

Choose differential_equations_18f46e29ae5df6d58883632ac9a73c6a3ea7e102.png satisfying differential_equations_4f10da36dba1d648328cd16426ccec3c75522348.png such that for all differential_equations_5c20ca8c9a26b6c01a7752f23e6ca64f2d9afed4.png. Such a choise is always possible, again because of the continuity of differential_equations_83202b174a58234b0c4164fcf4b60cbed4d4539a.png.

Now, consider any differential_equations_1507b6bfae935ba4e2acf157ddc83eab3f2835d5.png such that differential_equations_a5dc3cf6d40418ec0152e24baefd12b8fa98a976.png and thus differential_equations_080581ec6ee6470db71ecb76789ed745fc77398e.png and let differential_equations_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png be the resulting trajectory. Observe that differential_equations_1368515ccada53ca631750765ffed396bf244b8a.png is non-increasing, that is

differential_equations_9c3b23c9a782ab675ed9336c4ecf2445e07f9377.png

which results in differential_equations_93ad4e6da3ee411cc7a5e7d3cec4b389b33c2829.png.

We will now show that this implies that differential_equations_5ac231682360d5714c99341eac36b961dfdc5e67.png, for a trajectory defined such that differential_equations_e377f17dacc42c3533502099ffd0b418f1b29aa9.png and thus differential_equations_aa73b6fff085e1fc5920abef47eed17499a22199.png as described above.

Suppose there exists differential_equations_e3718360f6ee2d07be9d43498e4b14ec914da2f6.png such that differential_equations_31b19e9ab260d83bbe468d997c29c97f26a173c8.png, then since we're assuming differential_equations_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is such that

differential_equations_1a78842fc72d3bb09301d79d005e46fd4595de9e.png

as described above, then clearly we also have

differential_equations_320d6fe71aa57c7c2fcfd5f160ddbfe5fa4ccd0b.png

Further, by continuity (more specifically, the IVT) we must have that at an earlier time differential_equations_59a5ca3db1cf877644908496a93f9d505843ea3a.png we had

differential_equations_a811e08419945e48b7aabde8b6b366b6b7965c93.png

But since differential_equations_beda217b8e8f1608a87f660317a6bd8faa52fb7c.png, we clearly have

differential_equations_193aaeeffa73609c8a9b1a24ceac6199ed8891a1.png

due to differential_equations_27d7cf9930ceeacaf995d733cb6a66120acd977d.png. But the above implies that

differential_equations_70c64ba2fde52e707cab27eef4263c61d9f36f02.png

Which is a contradiction, since differential_equations_4ec31ed83730ca6bf0406849c15006f10258827b.png implying that differential_equations_b46674ebb304de2c8e2d80bf807ca21b88234dcd.png.

Now we prove the theorem for asymptotically stable solutions! As stated in the theorem, we're now assuming differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png to be locall negative-definite, that is,

differential_equations_8b1d2ecb06b4084b6f6741762411e1752ddd404e.png

We then need to show that

differential_equations_e2439c5bd6625aab02c592315a9b9453fdc190de.png

which by continuity of differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, implies that differential_equations_fbc973b0947c7ef9e68e47d354a27e754602ec3a.png.

Since differential_equations_1368515ccada53ca631750765ffed396bf244b8a.png is strictly decreasing and differential_equations_65405122793c61298669cbf0fdd92dda22209c25.png we know that

differential_equations_9e6710ce4ef84ab0d5e6e9faecd6e9b82ba46430.png

And we then want to show that differential_equations_3b08fff7e6020eed2fff92534ac3e92b01b1558e.png. We do this by /contradiction.

Suppose that differential_equations_cceff0a93740f9b1b736f7e926efed7d24be194b.png. Let the set differential_equations_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png be defined as

differential_equations_015ad30d4cac2744d96f89c8b56b01e41319c522.png

and let differential_equations_abbb90a076d29d9e3099535c94244d533cb0b8db.png be a ball inside differential_equations_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png of radius differential_equations_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png, that is,

differential_equations_6204124f00399975b28d94f0bfde0b8ee7fb7412.png

Suppose differential_equations_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is a trajectory of the system that starts at differential_equations_1507b6bfae935ba4e2acf157ddc83eab3f2835d5.png.

We know that differential_equations_1368515ccada53ca631750765ffed396bf244b8a.png is decreasing monotonically to differential_equations_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png and differential_equations_1ede60cb5ce879df28f4a41d89aaf7b809345b98.png for all differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png. Therefore, differential_equations_15a4f054f7f7d8050fdba141858e6b1b1be4cf91.png, since differential_equations_da95efe2fd623d8960e19cd249c8a6a36d0e3eda.png which is defined as all elements for which differential_equations_59555bb4f9a72054a49e10626ac83db4f0363fdc.png (and to drive the point home, we just established that the differential_equations_1ede60cb5ce879df28f4a41d89aaf7b809345b98.png).

In the first part of the proof, we established that

differential_equations_10eec635dc978f4f87d7c34527f30bb847abc67f.png

We can define the largest derivative of differential_equations_83202b174a58234b0c4164fcf4b60cbed4d4539a.png as

differential_equations_8e20f71427a0da24c331ea51d3e360007b26e993.png

Clearly, differential_equations_be6a9ebfeb8699f2adaf9041f74771911492f416.png since differential_equations_50ef420ddfed40d54f91633c09e99db0a9122f4c.png is locally negative-definite. Observe that,

differential_equations_ed93fad99232f2e1ac7f6662cbc67ab0fb387b2e.png

which implies that differential_equations_b27081795f5eb2fd02e5bf4ec09996227266f700.png, resulting in a contradiction established by the differential_equations_cceff0a93740f9b1b736f7e926efed7d24be194b.png, hence differential_equations_3b08fff7e6020eed2fff92534ac3e92b01b1558e.png.

Remember that in the last step where say suppose "there exists differential_equations_e3718360f6ee2d07be9d43498e4b14ec914da2f6.png such that differential_equations_e9f8cbe4c8cd9cf23f8677eb47a143a64c53879b.png" we've already assumed that the initial point of our trajectory, i.e. for differential_equations_1507b6bfae935ba4e2acf157ddc83eab3f2835d5.png, was within a distance differential_equations_5ba7f95faf4929ae04d1184137c7e6e2e6e9f707.png from the critical point differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png! Thus we would have to cross the "boundary" defined by differential_equations_5ba7f95faf4929ae04d1184137c7e6e2e6e9f707.png from differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png.

Therefore, intuitively, we can think of the Lyapunov function providing a "boundary" which the solution will not cross given it starts with this "boundary"!

Side-notes
  • When reading about this subject, often people will refer to Lyapunov stable or Lyapunov asymptotically stable , which is exactly the same as we define to be stable solutions.
Examples of finding Lyapunov functions
  • Quadratic

    The function

    differential_equations_66b2d3a289dc507168378fa8ddb12717568619a5.png

    is positive definite if and only if

    differential_equations_1a1d7b9477c6b256a88fcfcee7eba32b425cff6d.png

    and is negative definite if and only if

    differential_equations_80a08383139b86c6ab9a4c02ec9edf1dcadebae8.png

    • Example problem

      Show that the critical point differential_equations_8a89c460fad56a68a765f42cd59ad8c9016fa0b3.png of the autonomous system

      differential_equations_ddf4e287c109c792ca2a89c4d5383d70fa0c8a92.png

      is asymptotically stable.

      We try to construct a Liapunov function of the form

      differential_equations_ee759ca585861026e3b08c2baee49938e7f31d1a.png

      then

      differential_equations_36979f7f26ac94ade6bff24d28f2c37ebc8c8976.png

      thus,

      differential_equations_22a30669e738613022a63847ef31850f1e9e8302.png

      We observe that if we choose differential_equations_9c7865017fe33e0d5b8db92e57b70949ad58242d.png, and differential_equations_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and differential_equations_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png to be positive numbers, then differential_equations_952be1c5de6d8842ad4be2c1321809acfc41baae.png is negative-definite and differential_equations_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is positive-definite. Hence, the critical point differential_equations_8a89c460fad56a68a765f42cd59ad8c9016fa0b3.png is asymptotically stable.

Partial Differential Equations

Consider the differential equation

differential_equations_eaff398445cd39b876fc37adf1a163a17f9f141d.png

with the boundary conditions

differential_equations_0a5531729bb081c7027bd109ff59fd08038bbb37.png

We call this a two-point boundary value problem.

This is in contrast to what we're used to, which is initial value problems, i.e. where the restrictions are on the initial value of the differential equation, rather than the boundaries of the differential equations.

Eigenvalues and eigenfunctions

Consider the problem consisting of the differential equation

differential_equations_3a8429e3d7cbf57f6e437782bcd35112b52ad099.png

together with the boundary conditions

differential_equations_f6bb4be54b3e59f100e1877c6be19f30e8f79472.png

By extension of terminology from linear algebra, we call each nontrivial (non-zero) constant differential_equations_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png an eigenvalue and the corresponding nontrivial (not everywhere zero) solution differential_equations_eb83d466c7d035356e9f39998f357cee73da1e26.png an eigenfunction.

Specific functions

Heat equation

Models heat distribution in a row of length differential_equations_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png.

differential_equations_4e6f847121781af40f454722ebe04cc7f88ba1f5.png

with boundary conditions:

differential_equations_c8ae25c700a6b25d3fe99d0cd537b9672770630d.png

and initial conditions:

differential_equations_19fbf4eebe38e5cbb70d06a5d697105125a7e2bb.png

Wave equation

Models vertical displacement differential_equations_42683648b06176ce7644a8b4be60ac09f9660b5c.png wrt. horizontal position differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png and time differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png.

differential_equations_08e9005bafde86373e8996b407f431f28795a602.png

where differential_equations_7f1372a6855746a053b48e7b3be2681258d228df.png, differential_equations_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png being the tension in the string, and differential_equations_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png being the mass per unit length of the string material.

Reasonable to have the boundary conditions:

differential_equations_4d4f22e0826dce7e5598f3d208712620c0e8f4b2.png

i.e. the ends of the string are fixed.

And the initial conditions:

differential_equations_dc8b369355abf29c89f575395d9830b01b530541.png

i.e. we have some initial displacement differential_equations_9a6c9e4e5580d33233d25060fb4b1ca77fc8b57a.png and initial (vertical) velocity differential_equations_9f5048834ae702cb1a8486235e8105df13f0990f.png. To keep our boundary condtions wrt. differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png specified above consistent:

differential_equations_387055d1b918a7f1acb17866aee40ce14b42ea76.png

Laplace equation

differential_equations_74303a1b4128ac54874f5a079834a85fd6fe1532.png

or

differential_equations_739dc56e9074a472a1fea5f7754600cdd7114b84.png

Since we're dealing with multi-dimensional space, we have a couple of different ways to specify the boundary conditions:

  • Dirichlet problem : the boundary of the surface takes on specific values, i.e. we have a function differential_equations_0c8ea2668b18330373ba4f605802495cdd760064.png which specifies the values on the "edges" of the surface
  • Neumann problem : the values of the normal derivative are prescribed on the boundary

We don't have any initial conditions in this case, as there's no time-dependence.

  • Laplace's equation in cylindrical coordinates

    Let

    differential_equations_2e48da8f63e99f9d7a6383c43849817325442335.png

    Laplace's equation becomes:

    differential_equations_b7c22f36e81442ff9c5546a23d2c0c40d2dc9b57.png

    Consider BCs:

    differential_equations_a6fb6a589b674def08fc4fef600201bbd0392bcf.png

    Using separation of variables: differential_equations_37bb8b9d9487ff5890b8fa125ed4ac2bc9851f29.png

    differential_equations_2585d381d1a26b4ae229d1fec65b4deacf8ffa8d.png

    we can rewrite as:

    differential_equations_c29ea6ddca394b81f3b290a415b705a80458a964.png

    Using differential_equations_0cbec0094f508bef0ecf53d837842867042e0aab.png, differential_equations_a9b672e57a48c9d5cb39452ae267106dd80b18a9.png, hence

    differential_equations_a7495d34116a1d2290fc1a299fe481491b700121.png

    Thus, if differential_equations_75951d86f4a33c679a11f44eb5bb4aded1e6a19a.png, solutions are periodic.

    The radial equation can be written in the SL-form

    differential_equations_6ac0e267731cfbf7c689d92267432081cc94bed8.png

    Thus,

    • differential_equations_89d82d64d84359234ddb2b19715871db9737562a.png: vanish at the origin differential_equations_55bcb50ac4b0cc12e0164b5e1ee7f3bf00ccf210.png
    • differential_equations_125935100960a2fc5492b8131022de5df6566481.png: unbounded as differential_equations_9dda08de9aaa6df98f32a0dcdd88710bbcdba117.png

    Finally, making the change of variables differential_equations_de076cbd7881de53f0281f6ec51ab539324d4a87.png amd write differential_equations_22b43f38d99521f2406980ff682292a2bb4bfbe0.png, we have

    differential_equations_c7702de7d5c3634d8db57097dcd2be0d2232bf59.png

    Which is the Bessel equation, and we have the general solution:

    differential_equations_6b14409c3f9a52993c66f10a0b294f70e1e3b942.png

    Imposing the boundary conditions:

    • differential_equations_06e8746e38b95b54d0e9e1fb8aa94a829beec996.png
    • differential_equations_c27a9b8df783d23bc04911f731b65896d97c9fd7.png as differential_equations_9dda08de9aaa6df98f32a0dcdd88710bbcdba117.png needs to be bounded
    • differential_equations_8be74c56e635a64da76158122eee8345979a939d.png and differential_equations_b8aa5dee958c1fc48b0a8e0f15fe4e6a8e8be5de.png for differential_equations_f4a0f4c049ae60af7b3f2179cc36f3980b3fbf71.png
    • differential_equations_a2bdd1cf8009253af2e02561fd9e90d6ef0e81b6.png as differential_equations_9dda08de9aaa6df98f32a0dcdd88710bbcdba117.png (due to log)

    Hence, we require

    differential_equations_febecf2b854c823e8b58de335a666e3c7aaceb2f.png

    Plotting these Bessel functions, we conclude there exists a countable infinite number of eigenvalues.

    We now superpose the solutions:

    differential_equations_46198438ed6e12d8f6d5a10c3b711c58aee14c5c.png

    to write the general solution as

    differential_equations_e13525738e29194c9df82c4915d34ff9439f1550.png

    The constants differential_equations_ea7a18cd098aabf1cc0c93c2599c4d9361817785.png and differential_equations_19a8de9f4485618eeb24240e9e8433b318045676.png are found from the remaining BC,

    differential_equations_442347e4d94b3dec8ad042f2340747afe49e69dc.png

    by projection on differential_equations_177695a902b811f78f73521c34482973dc47fd54.png and differential_equations_43ea436c81bc179b809a9987185af5164157625b.png.

    Orthogonality: the Besseul functions differential_equations_307af92076f5f63a0d0b388ee9cd00f9a3156c5b.png satisfy

    differential_equations_b400010ab890dc5ee60fc725c7908838bb0832b5.png

    Which has the Sturm-Liouville form, with

    differential_equations_0f8f875615db7e32031a8fc1865364d5abcc7d19.png

    provided that

    differential_equations_17b8dfec6c3a421b729811b21071e14158eb6bb6.png

    The functions differential_equations_ff779d758ceb324025d2eb48b9345257d9ba36f5.png satisfy the above at differential_equations_70de07ad8847bfa1ca1a4390a59ffba269541537.png, where the contribution at differential_equations_70de07ad8847bfa1ca1a4390a59ffba269541537.png vanishes, and at differential_equations_55bcb50ac4b0cc12e0164b5e1ee7f3bf00ccf210.png we have

    differential_equations_c25a639d5ff23df274c55e566dc20695d6f14adc.png

    since differential_equations_9c815fd5cb9f98d37f3fdc835a49f525503780a2.png and differential_equations_009cfbec64270dd94da88bce30f970d7bc4c7ca1.png and differential_equations_a77550b1639133790210396638a4bc40c44530ee.png are bounded.

    We conclude

    differential_equations_3e2f7a994963e21dbdb9d17264d065982a7ee0a6.png

    for differential_equations_81ec62c3e41f8a8826e7381baab6a12ff1ee9030.png. We can therefore identify the constants differential_equations_ea7a18cd098aabf1cc0c93c2599c4d9361817785.png and differential_equations_19a8de9f4485618eeb24240e9e8433b318045676.png by projection:

    differential_equations_cf6e11bd3f0add0a42bd00cff299bae117dbee2a.png

Boundary Value Problems and Sturm-Liouville Theory

Notation

  • differential_equations_d6fbce90f35cb1f21cfbf2e812b20b6eaf896cea.png
  • differential_equations_401bbe60cf9d242728f47ef52fd3202134edb781.png are the ordered eigenvalues
  • differential_equations_f279aeafcd82663a5ed00c829ededcaa0fe93219.png are the corresponding normalized eigenfunctions
  • differential_equations_0778f317a86a9c79cba5698ac19dab3e6c24de89.png and equivalent for differential_equations_1485efbe0c5ba7f6e8ed02a90fff0707e9b3a081.png

Sturm-Liouville BVP

Equation of the form:

differential_equations_4253fb562da16e485e9931882cd733d82a80fc72.png

OR equivalently

differential_equations_93e7594aab0440c257c17387071b7f6be8ff9e6d.png

with boundary conditions:

differential_equations_8aa7c4827b2a6375e2f7c15895b53b1a21636d5b.png

where we the boundary value problem to be regular, i.e.:

  • differential_equations_d63400566f54e02cfbc1925b9931940bbe8ff549.png and differential_equations_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png are continuous on the interval differential_equations_5cb819dbdaa11557a460fe04a5ef95eede596daa.png
  • differential_equations_286b5e64c70d0d94e8beeea064394874f9047ddb.png and differential_equations_edf01ee8921138486ec8c643f167c27cc218d9fb.png for all differential_equations_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png

and the boundary conditions to be separated, i.e.:

  • each involves only on the of the boundary points
  • these are the most general boundary conditions you can have for a 2nd order differential equation
Lagrange's Identity

Let differential_equations_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png and differential_equations_d198c0d06b0da5525f9a966222601066509ac01d.png be functions having continous second derivatives on the interval differential_equations_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png, then

differential_equations_ce2f10d114f6f15eaf553a04d0a3b57315d5bc95.png

where differential_equations_98a487c2210fb4917ea727664189a56b0f26f7f6.png is given by

differential_equations_62e519b236b44180495340d5ab8f5fcf14c683cf.png

In the case of a Sturm-Liouville problem we have observe that the Lagrange's identity gives us

differential_equations_141c11a881d36d6ae0a1cc88128513be51b758c5.png

Which we can expand to

differential_equations_260b1f69006912ec6b3992ad5356cdf3faf29acd.png

Now, using the boundary conditions from the Sturm-Liouville problem, and assuming that differential_equations_1c2da0842a5fbba2b12ab8d296ebc991bda89488.png and differential_equations_d923e4c83e268e7739fc5cc35a0acdcf858be3f4.png, we get

differential_equations_729a1e9cf5cb2ebe1eab64ed78a15d3bf17f4421.png

And if either differential_equations_9c0a426bb3bf9ceb0446e8198dedfecf147217a0.png or differential_equations_b0802f9f73b53d337b2294f63a612a55abfcaa40.png are zero, differential_equations_2ba6526048ddd63af7c874686ad3c7394223c754.png or differential_equations_07f65f240d5089ee34dc70f382d17ac0455b65e4.png and the statement still holds.

I.e. the Lagrange's identity under the Sturm-Liouville boundary conditions equals zero.

If differential_equations_57b6c007ed1466df18ce19e323283b7e768f2846.png are solutions to a boundary value problem of the form def:sturm-lioville-problem, then

differential_equations_32b892544380f1cc5d5050e2a58907ef1ea7f13c.png

or in the form of an inner product

differential_equations_d470666ebe1f44bf1cf8256e001473fac1b27e47.png

If a singular boundary value problem of the form def:sturm-lioville-problem satisfies thm:lagranges-identity-sturm-liouville-boundary-conditions then we say the problem is self-adjoint. More specifically,

differential_equations_93e7594aab0440c257c17387071b7f6be8ff9e6d.png

for an n-th order operator

differential_equations_2effb758b34da63d52467bae1d1c088325d2a214.png

subject to differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png linear homogenous boundary conditions at the endpoints is self-adjoint provided that

differential_equations_f206ee69d2add1012b7ff8736eafb2ec9dcf4b54.png

E.g. for 4th order we can have

differential_equations_b39042f536e11e9f10ffc3f4e9aff72961d1dc15.png

plus suitable BCs.

Most results from the 2nd order problems extends to the n-th order problems.

Some theorems

All the eigenvalues of the Sturm-Liouville problem are real.

If differential_equations_c19a433021086bf40c3f51cd62063a8632e8c777.png and differential_equations_1a5e8cc664d575dc9783f32b7f6803f5647c444b.png are two eigenfunctions of the Sturm-Lioville problem corresponding to the eigevalues differential_equations_bae2e9d249d366de7d1509980dccd799ca664a12.png and differential_equations_7d8200d4f3c27a51eb92d7ee5006c860d88b02f6.png, respectively, and if differential_equations_2a16a375525c35958068783e57c9a69da372d43f.png, then

differential_equations_ab60b980fc3bc8be0f9089f06e5c67b0cf9086a7.png

That is, the eigenfunctions differential_equations_527e31e149bbe978c4a495b5f1bf7336381f2cbe.png are *orthogonal to each other wrt. the weight function differential_equations_a30ba58fa8728bfd457ccad04817f3889b92247f.png.

We note that differential_equations_c19a433021086bf40c3f51cd62063a8632e8c777.png and differential_equations_1a5e8cc664d575dc9783f32b7f6803f5647c444b.png, with differential_equations_2a16a375525c35958068783e57c9a69da372d43f.png, satisfy the differential equations

differential_equations_9374b0f4442df740ed3b49f3a09478ce45bd3479.png

and

differential_equations_8715dace5b88b9636d81c3afb1f9c01c9ea8c0f9.png

respectively. If we let differential_equations_561b4983a7751c4c9a5a29482c28c4f3696bbc42.png and differential_equations_a08f90b9dc672ffda46b28965a5ed1bd1aac59a1.png, and substitute differential_equations_c022385df836628854033357c458463b6cafcc76.png and differential_equations_ac87be758d87eeda533f88dd3ee8bb3c9c326b35.png into Lagrange's Identity with Sturm-Liouville boundary conditions, we get

differential_equations_8d9245cc603cba75957f9afd31f3a6349d9bc9c2.png

which implies that differential_equations_8397449a70e371aa464baef7a48a0c8b68e06b9b.png where differential_equations_a2699a803bdcdbb49c394529456fccb17dd66298.png represents the inner-product wrt. differential_equations_a30ba58fa8728bfd457ccad04817f3889b92247f.png.

The above theorem has further consequences that, if some function differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png satisfies is continuous on differential_equations_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png, then we can expand differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png as a linear combination of the eigenfunctions of the Sturm-Liouville equation!

Let differential_equations_f279aeafcd82663a5ed00c829ededcaa0fe93219.png be the normalized eigenfunctions of the Sturm-Liouville problem.

Let differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png and differential_equations_448c2f570fd2e984d07d7ec3e27a2507188cee52.png be piecewise continuous on differential_equations_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png. Then the series

differential_equations_f1636be1f05bed3b99aba30f564ea70bdd76ccb0.png

whose coefficients differential_equations_e91785338507cfe9adcff026711d4799cd823911.png are given by

differential_equations_7f34872bd021576ba5022a48ef444909b1444703.png

converges to differential_equations_2f5c849de522dfc699cf842549fbb37c253ac6e3.png at each point in the open interval differential_equations_4a3b22922580e4a653364bd8f594a09dc4a793fb.png.

The eigenvalues of the Sturm-Liouville problem are all simple; that is, to each eigenvalue there corresponds only one linearly independent eigenfunction.

Further, the eigenvalues form an infinite sequence and can be ordered according to increasing magnitude so that

differential_equations_fbf9fe4aae253a91aaaefa0c23651be6cbe3a476.png

Moreover, differential_equations_e7c30336eac2e7c5d76f192e6b74b14218070033.png as differential_equations_2531823f4e49c847dd9a34d97de0ea57802b96fa.png.

Non-homogenous Sturm-Liouville BVP

"Derivation"

Consider the BVP consisting of the nonhomogenous differential equation

differential_equations_cc66e73844a24a51bd516d1d983ad29963c7a359.png

where differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a given constant and differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png is a given function on differential_equations_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png, and the boundary conditions are as in homogeonous Sturm-Lioville.

To find a solution to this non-homogenous case we're going to start by obtaining the solution for the corresponding homogenous system, i.e.

differential_equations_93e7594aab0440c257c17387071b7f6be8ff9e6d.png

where we let differential_equations_401bbe60cf9d242728f47ef52fd3202134edb781.png be the eigenvalues and differential_equations_f279aeafcd82663a5ed00c829ededcaa0fe93219.png be the eigenfunctions of this differential equation. Suppose

differential_equations_b87ca6d84265c0a35b88489ebb610b1a64815eea.png

i.e. we write the solution as a linear combination of the eigenfunctions.

In the homogenous case we would now obtain the coefficients differential_equations_b71180177ccb0c9947452ca910e5b2eb203ff462.png by

differential_equations_6452eb98d9de347a3d58580a15d15dbd16028cbe.png

which we're allowed to do as a result of the orthogonality of the eigenfunctions wrt. differential_equations_a30ba58fa8728bfd457ccad04817f3889b92247f.png.

The problem here is that we actually don't know the eigenfunctions differential_equations_1a5e8cc664d575dc9783f32b7f6803f5647c444b.png yet, hence we need a different approach.

We now notice that differential_equations_d21892f3bdfae9ec08a78f3061484c467c1030ec.png always satisfies the boundary conditions of the problem, since each differential_equations_1a5e8cc664d575dc9783f32b7f6803f5647c444b.png does! Therefore we only need to deduce differential_equations_b71180177ccb0c9947452ca910e5b2eb203ff462.png such that the differential equation is also satisifed. We start by substituing our expansion of differential_equations_d21892f3bdfae9ec08a78f3061484c467c1030ec.png into the LHS of the differential equation

differential_equations_46124ef4d22d253b59786bfbaba1923418ddaecc.png

since differential_equations_0485f8033f0e59493f4e19afddae7d84faa8a4c0.png from the homogenous SL-problem, where we have assumed that we interchange the operations of summations and differentiation.

Observing that the weight function differential_equations_a30ba58fa8728bfd457ccad04817f3889b92247f.png occurs in all terms except differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png. We therefore decide to rewrite the nonhomogenous term differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png as differential_equations_b0138bc0d1f13d1910fbce8f6268f300512fea82.png, i.e.

differential_equations_2d5ed81cb295efd8d6924d977024ef3acae9c799.png

If the function differential_equations_1fc3d248090789ee3a757e62b10d1aa4ed48d84f.png satisfies the conditions in thm:boyce-11.2.4, we can expand it in the eigenfunctions:

differential_equations_603103d51bda3524ae27b98edba6cb41f6142303.png

where

differential_equations_711960f05618a2960ab2440a7ea9cf03d9835276.png

Now, substituting this back into our differential equation we get

differential_equations_eca946441f7a1e825feaa8cc46e55a1c5d4918ea.png

Dividing by differential_equations_a30ba58fa8728bfd457ccad04817f3889b92247f.png, we get

differential_equations_6597e73ebe3d7c56c5260b64bee28180eae0cefb.png

Collecting terms of the same differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png we get

differential_equations_7cd288ad773203403100ee02a1eac81754e9d7c7.png

Now, for this to be true for all differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png, we need each of the differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png terms to equal zero. To make our life super-simple, we assume differential_equations_1bc313e5dbf3f92564c1136c46c06d9d5a54d702.png for all differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, then

differential_equations_1fda2772d7c8e9759ed6a924ff28d8e53588f0e4.png

and thus,

differential_equations_fa09f3deb71f06f8637e43b5ff7ffa6c3ab2e39e.png

And remember, we already have an expression for differential_equations_f7ae591990d4b793df00916b56f53bdcf775dcc2.png and know differential_equations_7d8200d4f3c27a51eb92d7ee5006c860d88b02f6.png from the corresponding homogeonous problem.

If differential_equations_5226036447b9aa516a41b1479a400450691b0537.png for some differential_equations_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png then we have two possible cases:

  • differential_equations_dfb0599b738358597f16e375cc45eae140917b69.png and thus there exists no solution of the form we just described
  • differential_equations_f91bce91a1807e9f49820f50221e38eaeff52711.png in which case differential_equations_d2bf3c54bf2867302e91ea22da88396b790b8969.png, thus we can only solve the problem if differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png is orthogonal to differential_equations_c19a433021086bf40c3f51cd62063a8632e8c777.png; in this case we have an infinite number of solutions since the m-th coefficient can be arbitrary.
Summary

differential_equations_8940e6d630ed42133f5a1110438ea907e71b9b39.png

with boundary conditions:

differential_equations_f1a1bdfc1174092cff4b4d228bd2a5c77d419a80.png

Expanding differential_equations_ed4013c135118a227af7e0e9e48262c796a76549.png, which means that we can rewrite the diff. eqn. as

differential_equations_bfef0557d8260c6ebd057981baddabf5a740ac59.png

We have the solution

differential_equations_c1c43a5153bff9f95491f3b2724c38bfc7e9644d.png

where differential_equations_7d8200d4f3c27a51eb92d7ee5006c860d88b02f6.png and differential_equations_7f77ee094405a62deded4c50ee9c085aeb6d307d.png are the eigenvalues and eigenfunctions of the corresponding homogenous problem, and

differential_equations_ad23fde99e93547dae0b32fbd852878accf5e539.png

If differential_equations_5226036447b9aa516a41b1479a400450691b0537.png for some differential_equations_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png we have:

  • differential_equations_dfb0599b738358597f16e375cc45eae140917b69.png in which case there exist no solution of the form described above
  • differential_equations_f91bce91a1807e9f49820f50221e38eaeff52711.png i which case differential_equations_5b05e805fc2026f3ee54a9add49a0adb7298f48c.png; in this case we have an infinite number of solutions since the m-th coefficient can be arbitrary

Where we have made the following assumptions:

  • Can rewrite the nonhomogenous part as differential_equations_a11bf3bfa4832b88edf390790f09a88e60c472b3.png
  • Can expand differential_equations_0909593a4895f14df82e5e70ae5f5a2a6505280d.png using the eigenfunctions differential_equations_c19a433021086bf40c3f51cd62063a8632e8c777.png wrt. differential_equations_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png, which requires differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png and differential_equations_448c2f570fd2e984d07d7ec3e27a2507188cee52.png to be continuous on the domain differential_equations_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png

Inhomogenous BCs Sturm-Liouville BVP

Derivation (sort-of-ish)

Consider we have at our hands a Sturm-Liouville problem of the sort:

differential_equations_44a3a34b59e797708efce1817793ca0863dce31c.png

for any differential_equations_56e4b5321ccfd64064636df5a21266af5420817d.png, satisfying the boundary conditions

differential_equations_c0c888f37814c3edd35dc4406e446989ab96f658.png

This is just a specific Sturm-Liouville problem we will use to motivate how to handle these inhomogenous boundary-conditions.

Suppose we've already obtained the solutions for the above SL-problem using separation of variables, and ended up with:

differential_equations_30919959d4116723b5ee71c4de9960f6024faed1.png

as the eigenfunctions, with the general solution:

differential_equations_2289eb7cb68adc7963cb8ba91975e15a167fc995.png

where differential_equations_c350ded32087ca9a0650f8ddf6bc8460a93be43b.png such that we satisfy the differential_equations_e1be13a993eade54e1f45c6badeeb7b062746946.png BC given above.

Then, suddenly, the examinator desires you to solve the system for a slightly different set of BCs:

differential_equations_47f4f84572f5fd87c656b31930f8164235a7496d.png

What an arse, right?! Nonetheless, motivated by the potential of showing that you are not fooled by such simple trickeries, you assume a solution of the form:

differential_equations_f31a909d352e593a63ab29eb350fce2576dfc6ee.png

where differential_equations_42683648b06176ce7644a8b4be60ac09f9660b5c.png is just as you found previously for the homogenous BCs.

Eigenfunctions (i.e. (countable) infinite number of orthogonal solutions)) arise only in the case of homogenous BCs, as we then still have the Lagrange's identity being satisifed. For inhomogenous BCs, it's not satisfied, and we're not anymore working with a Sturm-Liouville problem.

Nonetheless, we're still solving differential equations, and so we still have the principle of superposition available to work with.

Why would you do that? Well, setting up the new BCs:

differential_equations_d1b13b996693ca2becaae77998d2c6af0753f01a.png

Now, we can then quickly observe that we have some quite major simplifications:

differential_equations_1af12d1510bd9525e9c9ece3652b1076b623d8cd.png

If we then go on to use separation of variables for differential_equations_df9f6f74baa7f6a97249d33851a42baaec9f943a.png too, we have

differential_equations_b90509e07b49473395e06ccd8cd1d89c8a8150cf.png

Substituting into the simplified BCs we just obtained:

differential_equations_f76a845ae52544fa0844f97b7150edded5d04f6f.png

Here it becomes quite apparent that this can only work if differential_equations_f80f737e71f0fb4f29a8eead08fdff09e50d8ba6.png, and if we simply include this constant factor into our differential_equations_27b325d2ef101fa003f053a5b70eaf816a3c1476.png, we're left with the satisfactory simple expressions:

differential_equations_36e1c686b581f5a92b59ac82ad2d14da3d00327f.png

This is neat and all, but we're not quuite ready to throw gang-signs in front of the examinator in celebration quite yet. Our expression for the additional differential_equations_df9f6f74baa7f6a97249d33851a42baaec9f943a.png has now been reduced to

differential_equations_3e0f48c2c7a04b6d860493c8cbe65b0b6887246c.png

Substituting this into the differential equation (as we still of course have to satisfy this), we get

differential_equations_472d3af8875b86fc90d6ce82a3f2ec30ebb9d2b6.png

where the t-dependent factor has vanished due to our previous reasoning (magic!). Recalling that differential_equations_f90679a03e0e3801603e2f1cc3e3f7157eb92e3a.png, the general solution is simply

differential_equations_cff372db257b56401a4c9c5c7ed6afb9bca8d67a.png

Substituting into the BCs from before:

differential_equations_83dabbd0afff1241c5d97737e24688a931e6566a.png

where the last BC differential_equations_3bdf026197401226b8b1e49a78a5317c803003b0.png gives us

differential_equations_fb410f3b07d0729fc4b432dbcf5bfeb27c5e7a16.png

thus,

differential_equations_920f363e54e5fa222470bf08aa6a97cf0bc65f27.png

Great! We still have to satisfy the initial-values, i.e. the BCs for differential_equations_9b50c3c5095e3def3ea0a38afe0602245b66e068.png. We observe that they have now become:

differential_equations_a905d031766d4cc7d22173166d3b610ff41f4edf.png

where the last t-dependent BC stays the same due to differential_equations_5c059063e50cae2046db956bb1df6e8602f7c1d4.png being independent of differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png. For the first BC, the implication arises from the fact that we cannot alter differential_equations_27b325d2ef101fa003f053a5b70eaf816a3c1476.png any further to accomodate the change to the complete solution, hence we alter differential_equations_42683648b06176ce7644a8b4be60ac09f9660b5c.png. With this alteration, we end up with the complete and final solution to the inhomogenous boundary-condition problem

differential_equations_65b6f30ae1b172cb8a3e7a4a4d67d5b9e301f72d.png

where

differential_equations_febe697cbb1760c2fc51cdfb01bfed2b695a3be1.png

At this point it's fine to throw gang-signs and walk out.

Observe that differential_equations_27b325d2ef101fa003f053a5b70eaf816a3c1476.png is kept outside of the sum. It's a tiny tid-bit that might be forgotten in all of this mess: we're adding a function to the general solution to the "complete" PDE, that is

differential_equations_799f00082cbd734f680418d78e8862de4b6f4386.png

NOT something like this differential_equations_927b489ec29969a7c0feadd2b1015ad6cf565160.png.

I sort of did that right away.. But quickly realized I was being, uhmm, not-so-clever. Though I guess we could actually do it if we knew differential_equations_27b325d2ef101fa003f053a5b70eaf816a3c1476.png to be square-integrable! Buuut..yeah, I was still being not-so-clever.

Example

Suppose we we're presented with a modified wave equation of the following form:

differential_equations_44a3a34b59e797708efce1817793ca0863dce31c.png

for any differential_equations_56e4b5321ccfd64064636df5a21266af5420817d.png, satisfying the boundary conditions

differential_equations_c0c888f37814c3edd35dc4406e446989ab96f658.png

and we found the general solution to be

differential_equations_e2e3dd8b53e0b004f77feda060afcc2185af4c73.png

where differential_equations_f7ae591990d4b793df00916b56f53bdcf775dcc2.png is as given above.

Now, one might wanted, what would the solution be if we suddenly decided on a set of non-homogenous boundary conditions :

differential_equations_d9d8c1a08bd5c81e691e4e87ac7c5e27478fb6a0.png

To deal with non-homogenous boundary conditions, we look for a time independent solution differential_equations_a9c6afd69bc17e1af478ccb27e876d4a0d0e2a40.png solving the boundary problem

differential_equations_197d964aee8494394e7b8912acc365cece3da0bb.png

Once differential_equations_a9c6afd69bc17e1af478ccb27e876d4a0d0e2a40.png is known, we determine the solution to the modified wave equation using

differential_equations_39e9594fdf495a66e9d9d7ca1e8a98ef59942508.png

where differential_equations_20cb10dba9353dd31e10b83163eacf47e251b840.png satisfies the same modified equation with different initial conditions, but homogenous boundary conditions. Indeed,

differential_equations_11b38cf06a88a96d5b93f855f539d503ced584c6.png

To determine differential_equations_a9c6afd69bc17e1af478ccb27e876d4a0d0e2a40.png, we solve the second order ODE with constant coefficients. Since differential_equations_f90679a03e0e3801603e2f1cc3e3f7157eb92e3a.png, its general solution is given by

differential_equations_210e314486f2890c1796a5c66324b19492e553c7.png

We fix the arbitrary constants using the given boundary conditions

differential_equations_49bda8543987d35dfed62bdded1fd363fdb6cf07.png

Thus, the general solution is given by

differential_equations_2d185320722b957de87ce06c408e48d2044528a9.png

By construction, the remaining differential_equations_20cb10dba9353dd31e10b83163eacf47e251b840.png is as in the previous section, but with the coefficients differential_equations_015c9ebfefb7ca2f436752dc29a81bdf990987dc.png satisfying

differential_equations_0d2ee675ddb78876351b4ef721dcfaa4df8de779.png

Thus, the very final solution is given by

differential_equations_8c00bb7c06aa67a7ca7c852a36fe63ac9c7c0c35.png

with differential_equations_f7ae591990d4b793df00916b56f53bdcf775dcc2.png as given above.

Singular Sturm-Liouville Boundary Value Problems

We use the term singluar Sturm-Liouville problem to refer to a certain class of boundary value problems for the differential equation

differential_equations_7ee5e81178ee91ced514505b1e092d40e864c560.png

in which the functions differential_equations_2505a59508d6c3e4890841ce6226f9f9c69584af.png and differential_equations_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png satisfy the conditions (same as in the "regular" Sturm-Liouville problem) on the open interval differential_equations_7065be0cde4b21555b0701e4670b64ccba83fa90.png, but at least one of the functions fails to satisfy them at one or both of the boundary points.

Discussion

Here we're curious about the following questions:

  1. What type of boundary conditions can be allowed in a singular Sturm-Liouville problem?
  2. What's the same as in the regular Sturm-Liouville problem?
    • Are the eigenvalues real?
    • Are the eigenfunctions orthogonal?
    • Can a given function be expanded as a series of eigenfunctions?

These questions are answered by studying Lagrange's identity again.

To be concrete we assume differential_equations_5c302d31faee1dc741905749ed4f6a8e7454408b.png is a singular point while differential_equations_49c1e4c835061e1643ceb67c6a764cbd29a8b696.png is not.

Since the boundary value problem is singular at differential_equations_5c302d31faee1dc741905749ed4f6a8e7454408b.png we choose differential_equations_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png and consider the improper integral

differential_equations_ed815a4a7e16e3d8fff825c33f57d4837927a949.png

If we assume the boundary conditions at differential_equations_49c1e4c835061e1643ceb67c6a764cbd29a8b696.png are still satisfied, then we end up with

differential_equations_1c87acf2ec07567199dfb64d44112033e34b1f4a.png

Taking the limit as differential_equations_d1b02fae817763a9940d653a2ee018c296e7869b.png :

differential_equations_a4f5773eb7f10a7b85852318d6b84493d7c0e2cb.png

Therefore, if we have

differential_equations_b6244f6c08fc10373360e5dca54510f80deeae0d.png

we have the same properties as we did for the regular Sturm-Liouville problem!

General

In fact, it turns out that the Sturm-Liouville operator

differential_equations_ad6aa333643f71f33c6529d613367c49d7092a6f.png

on the space of function satisfying the boundary conditions of the singular Sturm-Liouville problem differential_equations_14a40f189f2ce698341c03cd5c099a337431c49f.png, is a self-adjoint operator, i.e. it satisfies the Lagrange's identity on this space!

And this as we have seen indications for above a sufficient property for us to construct a space of solutions to the differential equation with these eigenfunctions as a basis.

A bit more information

These lecture notes provide a bit more general approach to the case of singular Sturm-Liouville problems: http://www.iitg.ernet.in/physics/fac/charu/courses/ph402/SturmLiouville.pdf.

Frobenius method

Frobenius method refers to the method where we assume the solution of a differential equation to be analytic, i.e. we can expand it as a power series:

differential_equations_68660463271876a2a441e9a691eccf9537230dee.png

Substituting this into the differential equation, we can quite often derive recurrence-relations by grouping powers of differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png and solving for differential_equations_9c17aa32c24ff4dbc420cd454dec844aef475372.png.

Honours Differential Equations

Equations / Theorems

Reduction of order

Go-to way

If we have some repeated roots of the characteristic equation differential_equations_e542db33247ebfc1248e2ea0753d5e26248ca1d6.png , then the general solution is

differential_equations_ca3847dbe2638ee951a30f5895dc28c2c28728de.png

where we've taken the linear combination of the "original" solution

differential_equations_995ebaef33a3b5ca04a4c82d49e0f246480646ad.png

and the one obtained from reduction of order

differential_equations_b95c769c6078072c508d56d3bbb5218ec0aa07e4.png

General

If we know a solution differential_equations_f883f7bad5c674a3ca46434782cf71b1eb0c8a8c.png to an ODE, we can reduce the order by 1, by assuming another solution differential_equations_4617d3bc7f9ce70a049a4251b1fd0740df6c1078.png of the form

differential_equations_b19719568e5adb5d8231170d3dc75e6a7e015684.png

where differential_equations_d318a5efb8e9c28dd4bb1fead2cc1b6c4684a13f.png is some arbitrary function which we can find from the linear system obtained from differential_equations_6d82e966445a565af1e609bffa0808e04ef6299b.png, differential_equations_ecb33750c58885f2901bcba49b6dbcf038c666f1.png and differential_equations_f59c9b77e206c4e18aa29642334dcdc6d8897e2f.png. Solving this system gives us

differential_equations_520ca8d4676c20b1cdeddf77258dcd8aff7bcb79.png

i.e. we multiply our first solution with a first-order polynomial to obtain a second solution.

Integrating factor

Assume differential_equations_84036252402b177a8086cfdc0d23b6cbfac4e282.png to be a solution, then in a 2nd order homogeneous ODE we get the

differential_equations_75cb77844039ee1f53d9b6eef8c36f38c8d1c8e9.png

Which means we can obtain a solution by simply solving the quadratic above.

This can also be performed for higher order homogeneous ODEs.

Repeated roots

What if we have a root of algebraic multiplicity greater than one?

If the algebraic multiplicity is 2, then we can use reduction of order and multiply the first solution for this root with the differential_equations_d318a5efb8e9c28dd4bb1fead2cc1b6c4684a13f.png.

Factorization of higher order polynomials (Long division of polynomials)

Good website with examples

Suppose we want to compute the following

differential_equations_ce0dcddcd901ac44a6edbb26f79a176f9dfc6f7c.png

  1. Divide the first highest order factor of the polynomial by the highest order factor of the divisor, i.e. divide differential_equations_ca9cd3cf4905a2f936943103a99611175c9e47ad.png by differential_equations_3ff5a42137e518a594ed75018b95267802a94702.png, which gives differential_equations_c39e1cf19a4f0afba04bdd4f84a61e69a664d52a.png
  2. Multiply the differential_equations_c39e1cf19a4f0afba04bdd4f84a61e69a664d52a.png from the above through by the divisor differential_equations_ad1d790d682783de3faa93d66cfacf6870595be1.png, thus giving us differential_equations_9a68a28293e0409cfd141066208f9d4f39608025.png
  3. We subtract this from the corresponding terms of the polynomial, i.e. differential_equations_535e365053a59731ecc148900c0044c26e653dad.png
  4. Divide this remainder by the highest order term in of the divisor again, and repeating this process until we cannot divide by the highest order term of the divisor.

Existence and uniqueness theorem

Consider the IVP

differential_equations_bdb35b505d68cef79ab468c3d4dd2852ce1428f1.png

where differential_equations_7225b076f6e6326f1636b11d1aad8de58bcc4761.png , differential_equations_f63749027365e29025a5cb867262c465d2de65bb.png, and differential_equations_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are continuous on an open interval differential_equations_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png that contains the point differential_equations_00c9a4d81fb363af64a1ce99ddcc267080f04ddd.png. There is exactly one solution differential_equations_c5f9a61f1964fc66bd51f39ae47fa9ee11a1d53a.png of this problem, and the solution exists throughout the interval differential_equations_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png.

Exact differential equation

Let the functions differential_equations_61371e4f672a0f550e1f94554a71ecbdc20f9ac0.png and differential_equations_83d08520fb13187a1057424f0dc3d87068c2da5e.png, where subscripts denote the partial derivatives, be continuous in the rectangular region differential_equations_ef808d4a314c26929cd2dba6306d3add6c64761e.png. Then

differential_equations_daee136c3efab8d774aadced22fcb02227893365.png

is an exact differential equation in differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png if and only if

differential_equations_82a6954f6c526cee89d01d89fe388f3eb58030f7.png

at each point of differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png. That is, there exists a function differential_equations_541a8541573e130dc310c688d0524f6e8fc0a5e3.png such that

differential_equations_68bb04539bff9fcd696127ea24924e8b5c4fb7f4.png

if and only if differential_equations_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and differential_equations_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png satisfy the equality above above.

Variation of Parameters

Suppose we have the n-th order linear differential equation

differential_equations_f103fe99a798216f81d52517845ba029872b604c.png

Suppose we have found the solutions to the corresponding homogenous diff. eqn.

differential_equations_2fcc97f31a974899c651901abe8ecc2e6e247921.png

With the method of variation of parameters we seek a particular solution of the form

differential_equations_cfe0839ef881ded0cb55e9485ab319a526f0d174.png

Since we have differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png functions differential_equations_933c36933a4f59cb3d5100e48049f9f9d406736a.png to determine we have to specify differential_equations_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png conditions.

One of these conditions is cleary that we need to satisfy the non-homongenous diff. eqn. above. Then the differential_equations_97e10175daca76010e298f70cce0826a16a6c021.png other conditions are chosen to make the computations as simple as possible.

Taking the first partial derivative wrt. differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png of differential_equations_858f50a429026dc5a18e911c0d6cfc2bc4bd18fe.png we get (using the product rule)

differential_equations_ff4e6fad3c249ba027bb1c3cd4bfbb81febdf872.png

We can hardly expect it to be simpler to determine differential_equations_858f50a429026dc5a18e911c0d6cfc2bc4bd18fe.png if we have to solve diff. eqns. of higher order than what we started out with; hence we try to surpress the terms that lead to higher derivatives of differential_equations_f42b50c6619721049cd9d758da6596ddcb761f22.png by imposing the following condition

differential_equations_aed3618552c1381943099efaba238055a06e78c3.png

which we can do since we're just looking for some arbitrary functions differential_equations_e5d7a7637b3d17e9a0a9d07aea011ca76af56ada.png. The expression for differential_equations_98a8bb3ff68141b85cc65aa358def884835dc13a.png then reduces to

differential_equations_a101a393276864856d2af4c8587cdbfe38b6f373.png

Continuing this process for the derivatives differential_equations_abfb326c0db01e6b0326f788cf4430551a4575a8.png we obtain our differential_equations_97e10175daca76010e298f70cce0826a16a6c021.png condtions:

differential_equations_1fbcebd6d54fe3cfb1adcf865e0a21e710961e01.png

giving us the expression for the m-th derivative of differential_equations_858f50a429026dc5a18e911c0d6cfc2bc4bd18fe.png to be

differential_equations_d5d00c3789dac726d871e3420e7649819990759c.png

Finally, imposing that differential_equations_74b70f16fbcd4161d22c21012510a32d689a7c27.png has to satisfy the original non-homogenous diff. eqn. we take the derivative of differential_equations_e0c9f49c215f297068d1df67fcbbf7ab430413ca.png and substitute back into the equation. Doing this, and grouping terms involving each of differential_equations_fe4fbcb8d7952b2fb6ba6a376dda9c1f28dce4f1.png together with their derivatives, most of the terms drop out due to differential_equations_fe4fbcb8d7952b2fb6ba6a376dda9c1f28dce4f1.png being a solution to the homogenous diff. eqn., yielding

differential_equations_79a1c420d9948b45971b08a8bc06f329fee39190.png

Together with the previous differential_equations_97e10175daca76010e298f70cce0826a16a6c021.png conditons we end up with a system of linear equations

differential_equations_431975d724466f78a4f3ce3bc591a0cf11916b3c.png

(note the differential_equations_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png at the end!).

The sufficient condition for the existence of a solution of the system of equations is that the determinant of coefficients is nonzero for each value of differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png. However, this is guaranteed since differential_equations_d6f3123e21d52b46d2454b67926e741948352821.png form a fundamental set of solutions for the homogenous eqn.

In fact, using Cramers rule we can write the solution of the system of equations in the form

differential_equations_c27debccda1c13b07fbdffcce43aa69643002589.png

where differential_equations_654b7fd5d40298e1fb9b8222fd2990a4991965b9.png is the determinant of obtained from differential_equations_f53cf2c3985c80987bd9707fd5a0b28594db886f.png by replacing the m-th column by differential_equations_eefd56391990a2f1856893cbcc51ce16c3e5deac.png. This gives us the particular solution

differential_equations_504cdbd7254db089ed0054bc908b5a6a8a3489d3.png

where differential_equations_0eca193b6f6cd17cb61e1065c28719e1d5237614.png is arbitrary.

Suppose you have a non-homogenous differential equation of the form

differential_equations_feb8cc85ed2aaf2d640ae2b484cf86ae1d1b3d9c.png

and we want to find the general solution, i.e. a lin. comb. of the solutions to the corresponding homogenous diff. eqn. and a particular solution. Suppose that we find the solution of the homogenous dif. eqn. to be

differential_equations_00d9635a47d97b9a3fa76251386af8ab4d3f6db5.png

The idea in the method of variation of parameters is that we replace the constants differential_equations_9d332d4fed2c94ad223b2444050cb112714acc4a.png and differential_equations_553d82fcac5866479444a946a2d8735efd290fab.png in the homogenous solution with functions differential_equations_6d0964399be7f527570870c475b18dd2c39d3d15.png and differential_equations_adf64f433c00ec2abffd4de38c2eebf14a588808.png which we want to produce solutions to the non-homogenous diff. eqn. that are independent to the solutions obtained for the homogenous diff. eqn.

Substituting differential_equations_84f613b0aae7ab3bab18697cee672054cc616023.png into the non-homogenous diff. eqn. we eventually get

differential_equations_34d2d9a092d3c467eee2c4f4bdb1fb67f20110cb.png

which we can solve to obtain our general solution for the non-homogenous diff. eqn.

If the functions differential_equations_2505a59508d6c3e4890841ce6226f9f9c69584af.png and differential_equations_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are continuous on an open interval differential_equations_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png, and if the functions differential_equations_f883f7bad5c674a3ca46434782cf71b1eb0c8a8c.png and differential_equations_4617d3bc7f9ce70a049a4251b1fd0740df6c1078.png are a fundamental set of solutions of the homogenous differential equation corresponding to the non-homogenous differential equation of the form

differential_equations_feb8cc85ed2aaf2d640ae2b484cf86ae1d1b3d9c.png

then a particular solution is

differential_equations_bc23c0f5c5a0bbf279d9fb7d6b7e87ecbe5a7152.png

where differential_equations_0eca193b6f6cd17cb61e1065c28719e1d5237614.png is any conventional chosen point in differential_equations_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png. The general solution is

differential_equations_299f39870d17c6795f16a534c0e7508983455475.png

Parseval's Theorem

The square norm differential_equations_482e9c5cb489a244e7d0b833094161d61bee6c02.png of a periodic function with convergent Fourier series

differential_equations_cce5bd2f2ca622296e5e28c1b14bb09b8ef7a34d.png

satisfies

differential_equations_d484cbcaf05f1067269555cbfbadc0ab2be0c698.png

By definition and linearity,

differential_equations_866dae9b555208dea7d9e8a04db8d38aac4cc031.png

For the last equality, we just need to remember that

differential_equations_fbab906bfce0c9326cd24a11a37d9f0f12df5118.png

Indeed,

differential_equations_b14643b9b222f6a682140cfffc00ee227c7d6ef9.png

We also have a Parseval's Identity for Sturm-Liouville problems:

differential_equations_73bb007d0a655bebad0b0c39d783845654ea765f.png

where the set differential_equations_000a0e9fc881df8dd7b447d0b5b1f8d0ee5580c8.png is determined as

differential_equations_78cbd218579a34d744a3764d90246cbd25fd1301.png

and differential_equations_1a5e8cc664d575dc9783f32b7f6803f5647c444b.png are the corresponding eigenfunctions of the SL-problem.

The proof is a simple use of the orthogonality of the eigenfunctions.

Binomial Theorem

Let differential_equations_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png be a real number and differential_equations_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png a postive integer. Further, define

differential_equations_ba0f06c7c30000733716602efc3f292280930347.png

Then, if differential_equations_9814019390683d0c946fcfeba0a8d88f8a65cad1.png,

differential_equations_3ec83863f640e318640c2765f59ad63879fc781c.png

which converges for differential_equations_3a4a3cea82b79f895ebe3e8eeb075b2021481742.png.

This is due to the Mclaurin expansion (Taylor expansion about differential_equations_5c302d31faee1dc741905749ed4f6a8e7454408b.png).

Definitions

Words

bifurication
the appearance of a new solution at a certain parameter value

Wronskian

For a set of solutions differential_equations_8246cc51a8a625655aa03990fa46e8a9fee626f7.png, we have

differential_equations_3705994461be3790862ed73602de6a8e1c7e2250.png

i.e. the Wronskian is just the determinant of the solutions to the ODE, which we can then use to tell whether or not the solutions are linearly independent, i.e. if differential_equations_55ea0f38a4cc7cd35ad20e46e73b7025c47bbdc0.png we have lin. indep. solutions and thus a unique solution for all initial conditions.

Why?

If you look at the Wronskian again, it's the determinant of the matrix representing the system of linear equations used to solve for a specific vector of initial conditions (if you set the matrix we're taking the determinant of above equal to some differential_equations_36d65c59bf612962452dc8928bcab18f81864d34.png which is the initial conditions).

The determinant of this system of linear equations then tells if it's possible to assign coefficients to these different solutions differential_equations_fe4fbcb8d7952b2fb6ba6a376dda9c1f28dce4f1.png s.t. we can satisfy any initial conditions, or more specific, there exists a unique solution for each initial condition!

  • if differential_equations_462710a4d41195e4d61375a458b07d7a325da8ea.png then the system of linear equations is an invertible matrix, and thus we have a unique solution for each differential_equations_36d65c59bf612962452dc8928bcab18f81864d34.png
  • if differential_equations_0dc4eb882a5e684ccd61a4c4cc29e5972909f460.png we do NOT have a unique solution for each initial condition differential_equations_36d65c59bf612962452dc8928bcab18f81864d34.png

Thus we're providing an answer to the question: do the linear combination of differential_equations_c2455be58ab92e681ce74548eef1180269bae96b.png include all possible equations?

Properties
  1. If differential_equations_462710a4d41195e4d61375a458b07d7a325da8ea.png then differential_equations_c9969ebdaf04e319031701f195cc43b59875659b.png for all differential_equations_e8a2d7b1cb93b4c595771140d7989477bd42b288.png
  2. Any solution differential_equations_6d82e966445a565af1e609bffa0808e04ef6299b.png can be written as a lin. comb. of any fundamental set of solutions differential_equations_a372e9008c2d77a7429cc53447deffb33e16d005.png
  3. The solutions differential_equations_a372e9008c2d77a7429cc53447deffb33e16d005.png form a fundamental set iff they are linearly independent
Proof of property 1

If differential_equations_51a5f0b6f71f57c78a0df0fb72e29a08389bf1e6.png somewhere, differential_equations_61c028b1be074943878203627f1d124db866d961.png everywhere for differential_equations_e8a2d7b1cb93b4c595771140d7989477bd42b288.png

Assume differential_equations_ec63e06eead2e90501a31487b4c01ab7d13435c8.png at differential_equations_97d2f277da4b97dd8e2a7ac1c3a61b3a56ff120f.png

Consider

differential_equations_176573cad5a8af9e3e75657ffabc1adfab7f273f.png

which can be written

differential_equations_e1387cefe600d63eca1eeb7f2cee469768a94e2a.png

with differential_equations_4b39e5137ab3ba7298f9b37db3293e04e00a97cb.png

Hence, differential_equations_82a75ad487d333c562281eb991e82a29195448e0.png such that (*) holds.

Thus, we can construct differential_equations_3048e556da95ec808885cb30e5c807583c3b65da.png which satisfies the ODE at differential_equations_43d2c6751499c0d42dc3a994ab21dc443f75b4b2.png

Proof of property 2

We want to show that differential_equations_8a787178ef68d53fac8592abb7d0ed1ba9c153ff.png can satisfy any

differential_equations_b32ccd55c7f70c12bcc1ddf74f79e03cfa3894ed.png

Linearly independent → differential_equations_2f61c4aa78afabe044ca73ee253b68933904b593.png and since differential_equations_8aedf5c3c5fb266ebb7db4ef94738fa90b54850a.png, by property 1 we can solve it.

Homogenenous n-th order linear ODEs

differential_equations_a3ad021421dc46af7531ffd232ba81e3bb8c74e0.png

Solutions differential_equations_21ea97fe04dc6bf896be13f3cf5419bc40c4b5f5.png to homogeneous ODEs forms a vector space

Non-homogeneous n-th order linear ODEs

differential_equations_b52fc6dd2723756d49d15e732a6ca06c14b7e1c3.png

Solution
  1. Solve the corresponding homogeneous linear ODE
  2. Find special case solution for the non-homogeneous
  3. General solution is then the linear combination of these
  • Simple case

    In the simple case of one variable:

    differential_equations_a5db8a064f1693c860a5bf6a6eef86e4fb0de84f.png

    where differential_equations_f883f7bad5c674a3ca46434782cf71b1eb0c8a8c.png and differential_equations_4617d3bc7f9ce70a049a4251b1fd0740df6c1078.png are a fundamental set of solutions of the corresponding homogenous equation, and differential_equations_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png is some specific solution to the non-homogenous equation.

Power series

differential_equations_22d789c7c08c846973995fd5d40ac8b726aceb0b.png

Laplace transform

An integral transform is a relation of the form

differential_equations_ce874148e8d77fd662c559d92bd06fcb36d98971.png

where differential_equations_a59653e92987e9c29875237ec8748ec9d1b03628.png is a given function, called the kernel of the transformation, and the limits of integration differential_equations_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png and differential_equations_81c5f489718d4ff0ca6962103920c3133c0daea4.png are also given.

Suppose that

  1. differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png is piecewise continuous on the interval differential_equations_0d426335d9ed1a7524fc1ce1df9bf55fd962bc98.png for any positive differential_equations_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.
  2. differential_equations_9f1806636d0b90e45b9650c30ac66e651c973f03.png when differential_equations_4b9150c26fbe4cf33c867e9b6366724c46c9fb9c.png. In this inequality, differential_equations_1641d18cc980f8db14cdff95d7417a8526eef446.png, differential_equations_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png, and differential_equations_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png are real constants, differential_equations_1641d18cc980f8db14cdff95d7417a8526eef446.png and differential_equations_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png necessarily positive.

Then the Laplace transform of differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png, denoted differential_equations_4b5cdc7d2ec351a51f2456ff5c4632b3e55f8bcf.png or by differential_equations_09a6a6e1512acb7b09892fb1fd9bbade4b209622.png, is defined

differential_equations_e425e172e07ecf5f7c74146f397e76a85ba6a17a.png

and exists for differential_equations_be26d829a8ee471cb885bbf922a008db84835356.png.

Under these restrictions a function differential_equations_9a6c9e4e5580d33233d25060fb4b1ca77fc8b57a.png which increases faster than differential_equations_db96bd1f613a72f0ac76ad5da1bb508f1dca016d.png is does not converge and thus the Laplace transform is not defined for this differential_equations_9a6c9e4e5580d33233d25060fb4b1ca77fc8b57a.png.

Properties
  • Identities

    Let differential_equations_9a925b317e0c9807069729f2e8463b10bfba4123.png, for differential_equations_363590f035590045c460a63e663c598871be89ac.png

    • S-shift

      differential_equations_788dd4fea52a9d2288418231555eb1b9b3a3bcd3.png

    • X-shift

      If differential_equations_3f3d701019ffdfe83104f303651c570bddafbdbe.png and differential_equations_6a3483facd6a491101c75a7860766f9e133e5617.png for differential_equations_afa8f0ea28736ef13aa9ce3354e7b18ef6a776b1.png:

      differential_equations_68ab61de7be6aa03b08c62c9d2b98066220761c3.png

    • S-derivative

      differential_equations_29d06bd70948044f716435453293b43619796f12.png

      or more generally,

      differential_equations_7e6905d5819e3bdf127ebe3f1f0f9e6034c55a71.png

    • Scaling

      If differential_equations_cceff0a93740f9b1b736f7e926efed7d24be194b.png:

      differential_equations_14570cda35a2476c8cd43e537f8f3be9aace949f.png

      differential_equations_14fae82bf1f647a09af4b970463ce2c109d0ebe5.png

  • Linear operator

    The Laplace transform is a linear operator:

    differential_equations_19c02bd8bfe9133ef72b6492a8079a48689b3579.png

  • Uniqueness of transform

    We can use the Laplace transform to solve a differential equation, and by

  • Inverse is linear operator

    Frequently, a Laplace transform differential_equations_09a6a6e1512acb7b09892fb1fd9bbade4b209622.png is expressible as a sum of serveral terms

    differential_equations_7cbed8a2cd3e0b1b6c5f5aec923f9d5d7417f6d1.png

    Suppose that differential_equations_7547288622fa3d548c67b447c0f189e4938da238.png. Then the function

    differential_equations_7006453b989b7986d8271c567e29f5979c2c039c.png

    has the Laplace transform differential_equations_09a6a6e1512acb7b09892fb1fd9bbade4b209622.png. By the uniqueness property stated previously, no other continuous function differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png has the same transform. Thus

    differential_equations_4c8af420e52745664a72340b304fd721b4c19257.png

    that is, the inverse Laplace transform is also a linear operator.

Solution to IVP

Suppose that functions differential_equations_c58e7e11ee7ac26d07cbe510e63a5566b519961c.png are continuous and that differential_equations_e0ca8447b8483477eed7c3540ddf6f24b13d5001.png is piecewise continuous on any interval differential_equations_0d426335d9ed1a7524fc1ce1df9bf55fd962bc98.png. Suppose further that

differential_equations_9c901465dde6b9ef50a3aebd1f99433ee71498ef.png

Then differential_equations_4b5cdc7d2ec351a51f2456ff5c4632b3e55f8bcf.png exists for differential_equations_cec2b83a080a472a932aeb42b9c57726ae61c8aa.png and is given by

differential_equations_47997753106a07ee539619eaa7b31a3fedeebfcc.png

We only prove it for the first derivative case, since this can then be expanded for the nth order derivative.

Consider the integral

differential_equations_ae2b84408b65587b332d628821ad4d0b35091253.png

whose limit as differential_equations_c9c5f32124fee8368921ea51c544131c14745aa8.png, if it exists, is the Laplace transform of differential_equations_448c2f570fd2e984d07d7ec3e27a2507188cee52.png.

Suppose differential_equations_448c2f570fd2e984d07d7ec3e27a2507188cee52.png is piecewise continuous, we can partition the interval and write the integral as follows

differential_equations_577a2850b1260197c2b2abf0f1716816f217bf64.png

Integrating each term on the RHS by parts yields

differential_equations_0b685f37c35d01b7499fe5350d01cd061f455b12.png

since differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous, the contributions of the integrated terms at differential_equations_6428611429029caa5e5bcb1f4f6037b6a4ed12f0.png cancel, and we can combine the integrals into a single integral, again due to continuity:

differential_equations_75603b9d4e10b8969b2a09bd0f4166002ae49d29.png

Now finally letting differential_equations_c9c5f32124fee8368921ea51c544131c14745aa8.png (since that's what we do in the Laplace tranform ) we note that

differential_equations_75ef3fb112db7c0268a14bcc87ce0bd587eb52b0.png

And since differential_equations_151e53bd9839bfb4ae6094998d8d930504cf3d2c.png, we have differential_equations_d79d283f1cbc3804979229ea7594f50b4aa7c818.png ; consequently

differential_equations_93500cd35bfc5ede1f07ae5e9e37f71cb88309aa.png

Finally giving us

differential_equations_b56e53652b9d3bfa339fdf4ba856e18a2f42f62d.png

Typical ones
Examples:

Fourier Series

Given a periodic function differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png with period differential_equations_f9d791a57dd9f478d23885cf29809f43a4c4ea43.png, it can be expressed as a Fourier series

differential_equations_a6b7a8c84d8540922f1777f7860c7901b6e5c979.png

where the coefficients are given by

differential_equations_6b88c78a298bb85b3954963b853c04f6f4bdb142.png

Suppose differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png and differential_equations_562e4c81813491489393ccdb3875f72dbcae156d.png are piecewise continuous for differential_equations_e0fc5316abdd15c109fab3cfb7e90142528f1ae2.png.

Suppose differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png is defined outside the interval so that it is periodic with period differential_equations_f9d791a57dd9f478d23885cf29809f43a4c4ea43.png. Then the Fourier series of differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png converges to differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png where differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png is continuous and to differential_equations_9891336fdd11ca38ff54c3074d0b38a2dabce613.png where differential_equations_b4625aa4535b6123d93c130d4dc4772a395916cd.png is discontinuous.

Heavside function

differential_equations_55d4013e567b8d591f6ac8050c918dd868aa2ec7.png

Which has a discontinuity at differential_equations_60eaaa63eddbb53b1058efb80d4eaca4d1d6e4da.png.

Convolution integral

If differential_equations_9a6c9e4e5580d33233d25060fb4b1ca77fc8b57a.png and differential_equations_9f5048834ae702cb1a8486235e8105df13f0990f.png are piecewise continuous functions on differential_equations_dfba454ddab9caf4e291e5dbf30de4a218c340d6.png then the convolution integral of differential_equations_9a6c9e4e5580d33233d25060fb4b1ca77fc8b57a.png and differential_equations_9f5048834ae702cb1a8486235e8105df13f0990f.png is,

differential_equations_08ae7fc7e63e7b50526a71ef586fd2433f2dfe6f.png

Laplace transform

differential_equations_573e43603e65fd84f6bf57e6cb77d4d8846d2c89.png

Phase plane

A phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coodinate plane with axes being the values of the two state variables, e.g. differential_equations_bcaca16349a57fa4db86decededc11f33eed0a02.png.

It's a two-dimensional case of the general n-dimensional phase space .

Trajectory

A trajectory differential_equations_c73cd3b2b54e5cf43171325d44788fe6bc949dfb.png of a system

differential_equations_cf00705b8c1569222c04c4c7688910828a3f7d8b.png

simply refers to a solution for some specific initial conditions differential_equations_cd8e20dba90a9009c6135d2413dce5bad16f4099.png, i.e.

differential_equations_07b65e38b1df4cabeaf9a4d7a56371c1b1eefb95.png

Critical points

Points corresponding to constant solutions or equilibrium solutions, of the following equation

differential_equations_e9e48d1c8919fe516f943ab9bac57cd97b939dae.png

are often called critical points.

That is, points differential_equations_bae4bae48eb3ea91a3404b3c1514d97719620616.png such that

differential_equations_91439815382c5066b435585b2ee8ea956ddc714f.png

There always exists a coordinate transformation differential_equations_cf6cab6669cf05119f0df17ebbc823ec617bc736.png, such that

differential_equations_56803d34087d24859f4394e73819831ebce8d69b.png

such that the new system

differential_equations_a3c885e80194d3f215956881406e74c587a5284f.png

has an equilibrium solution or critical point at the origin.

Special differential equations / solutions

Legendre equation

The Legendre equation is given below:

differential_equations_cdadcf7984534ae105c7bde3b171ed64773f6254.png

We only need to consider differential_equations_7334c46cf40e998ed8bfc45332661441f2b5fbee.png since if differential_equations_d24af6d27621a513ef49cd8abafb5d270842e0e7.png we can reparametrize the equation to be of the form above anyways.

In the case where we assume differential_equations_92d68125ba8e8e87d5178d9ee7c70f29bcabdac8.png, we note that for even differential_equations_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png differential_equations_f883f7bad5c674a3ca46434782cf71b1eb0c8a8c.png becomes a terminating series, rather than an infinite series, and if differential_equations_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png is odd differential_equations_4617d3bc7f9ce70a049a4251b1fd0740df6c1078.png is a terminating series, i.e. in both cases we get a polynomial of order differential_equations_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png !

These polymomials corresponding to a solution of the Legendre equation for a specific integer differential_equations_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png we call a Legendre Polynomial, denoted differential_equations_a597b064eaeb6d43affff9f0f49e9290f6f49d07.png.

differential_equations_7fbe206c3d13489afc178df04dd5e9d6a68a483a.png

Or the general Legendre equation :

differential_equations_c2a6f4e0a316c2747efad6173dbeb1fac7b01622.png

Which gives rise to the polynomials defined

differential_equations_1fc7ce7fcc8871b5db76b267b510a1722d29e34e.png

where differential_equations_a597b064eaeb6d43affff9f0f49e9290f6f49d07.png are the corresponding Legendre polynomial.

Moreover, using Rodrigues formula

differential_equations_d358e91ce0ca34bd6ea981384a16bf44da99462e.png

If we let differential_equations_6bf7086500f484bb0ba9f155229f3b7a06741c7b.png we can use the relationship between differential_equations_4bd53f22b005e9ab0cdd6af5144bc1439d21df39.png

differential_equations_db1bccc8fc603d0f502b973fe2cb59352707df69.png

to obtain the coefficient

differential_equations_ad1013ca44db48d94acae960c08de37409cb9fdf.png

Series Solutions

The sum of the even terms is given by:

differential_equations_dc1b6eb1a8f82d0f87c0e69793792ce0cd451dd5.png

and the sum of the odd terms are given by:

differential_equations_830c214f527065ae38f59aba6582a348e8b99936.png

TODO Legendre polynomials and Laplace equation using spherical coordinates

Van der Pol oscillator

differential_equations_003610844459a2eb1bd07ce9a52edc72965e2927.png

where differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png is the position coordinate, which is a function of time differential_equations_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, and differential_equations_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a scalar parameter indicating the nonlinearity and the strength of the damping.

Hermite polynomials

We are talking about the physicsts' Hermite polynomials given by

differential_equations_e2fef64100b54ff3ea69a09a0b134aa9f4b655ac.png

These are a orthonormal polynomial sequence, and they arise from the differential equation

differential_equations_bcd935d4d71211f9c504f79a1a5748d9dda8d20a.png

These polynomials are orthogonal wrt. the weight function differential_equations_573be2c36e21e1928293a4991be1c6f01bc944f8.png, i.e. we have

differential_equations_43850e9d6040907a0451b9c187e1e15161d49ae7.png

Furthermore, the Hermite polynomials form an orthogonal basis of a the Hilbert space of functions satisfying

differential_equations_d19bc18b52266407661ef19d5f74623ba700239f.png

in which the inner product is given by the integral including the Gaussian inner product defined previously:

differential_equations_1be5d86ffc2ad987ac291fdb13acf1d8fed6b9cf.png

Spherical harmonics

In this section we're looking to tackle the spherical harmonics once and for all.

If we have a look at the Laplace equation in spherical coordinates, we have

differential_equations_02202bcbf3e3a68f76fb4c662bb37acffa3cd9dd.png

If we then use separation of variables, and consider solutions of the form:

differential_equations_ed5045f60770f8122d68bee093bbf019f344f2d4.png

We get the following differential equations:

differential_equations_7edd6d7aec0c3c98b83156597e5bff48c62f77f7.png

Further, assuming differential_equations_fbfa6649835e25563dc0fd495c3397dc7d9086c4.png, we can rewrite the second differential equation as

differential_equations_42be807be20545cabe5ef4bb122165b3fe3e5a8e.png

for some number differential_equations_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png.

Solving for differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png

First we take a look at the differential equation involving differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png,

differential_equations_befa00a2ef00ce3dfdc70e22be55ef6551bb47fb.png

Which we can rewrite as

differential_equations_07ceb28e60263ec82a95af4fca5af89d318204b1.png

making the substitution

differential_equations_04b340c6f32206536715444f679a8c617a20eca7.png

This is fine, since in spherical coordinates we have differential_equations_efd774c6d9ede5fa30c7449ef695f33aaf0c64f2.png, i.e. differential_equations_218ef348a050bf78dfc4a49e5fdb605e31395cba.png is always defined.

Thus,

differential_equations_eafacdce569efd439e7f5bb147f864c6592d81a2.png

Substituting back into the differential equation, we get

differential_equations_382b0d25cd74b49ad85dc9b05bb14256144203e2.png

which has the general solution

differential_equations_082a38f3f4cd61defd12e712453b2d990064d20c.png

where

differential_equations_2da1e7efa27b5271b4567803e13b620c1b98a65f.png

solves

differential_equations_d126a7f0c12b72a5e91d1bbf0c8aaa09249d93f2.png

or rather,

differential_equations_2c6fa65193997f619e55eae926b87ae2a3cd230c.png

RECOGNIZE THIS YO?! How about trying to substitute this into the differential equation involving differential_equations_07aa11d29578308bc0642bd6f5d3e959f8d85582.png?

Solving for differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png

The differential equation with differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png can be written

differential_equations_c91aa5f4bd147da6f1676937ffaca9ad60220d1e.png

which has the solution

differential_equations_64ca2bebe1d756c0a9359c55c9572ed8aa97e8c3.png

Further, we have the boundary condition differential_equations_1cbaab2761a681fda45324e6ba33677e5130553e.png, which implies differential_equations_f56b556b5aef28b769c52c4c0bf2d3213e67eb2e.png.

Solving for differential_equations_07aa11d29578308bc0642bd6f5d3e959f8d85582.png

We have the the ODE

differential_equations_5845f97f1ae188a82d8cec7be25ab95bd31a2f62.png

which we can write out to get

differential_equations_70213554a0ac64769c181f70df60cd11f1036291.png

Making the substitution differential_equations_82f81c310a6e5ae8c00dde21a4193ede0e5a8a2b.png we have

differential_equations_6e8ea24f94dd697f5db16ad729b2444405cd89df.png

And,

differential_equations_dba773086a43f64c24b951dde192e4912cde77cc.png

Substituting back into the ODE from above:

differential_equations_79748c0e935f7a960a077c0e721188d0181d79ff.png

Observing that

differential_equations_316c6fd5b685ddf0d39d98229d91e84b05635784.png

We can write the above equation as

differential_equations_74237e7fa71c6ba7bcfc955ef8b3097c73c3627b.png

Whiiiich, we recognize as the general Legendre differential equation!

differential_equations_ef0125281ae9396ce75854e9f4bb608e302179ca.png

where we're only lacking the differential_equations_c0e7e486574eda3063c12f9f7e267aa761dc1ee4.png, for which we have to turn to the differential equation involving differential_equations_2475f047fcda96d98d2d69c2383c13cb8823b2f0.png.

With differential_equations_c0e7e486574eda3063c12f9f7e267aa761dc1ee4.png obtained from Solving for differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png, we can rewrite the differential equation involving differential_equations_07aa11d29578308bc0642bd6f5d3e959f8d85582.png as

differential_equations_c6e274a56961f99521f3e94318ea2d696b943730.png

For which we know that the solutions are the corresponding associated Legendré polynomials

differential_equations_7b2317780311baafe208fb923e9114d0547fff91.png

which are obtained by considering the series solution of the equation above, i.e. assuming:

differential_equations_cd2ad07b1227a204e3676dd3467883a420950515.png

where differential_equations_7a84c9a383f9772338016d101ccc096be06af784.png is as above. Substituting into the differential equation above, and finding the recurrence-relation for differential_equations_9c17aa32c24ff4dbc420cd454dec844aef475372.png we recover the Legendre polynomials mentioned above.

Bringing differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png and differential_equations_07aa11d29578308bc0642bd6f5d3e959f8d85582.png together

Now, finally bringing this together with the solution obtained for differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png, we have the spherical harmonics

differential_equations_8540f7b7bd9c3358b5cc980927e2a234f5fa08ff.png

where differential_equations_f039d697b15f329435f258e1a814b7b94bed986d.png is simply the normalization constant to such that the eigenfunctions form a basis, which turns out to be

differential_equations_96babeacf134e75fb0d79c4787819567611ddf09.png

Thus, the final expression for the spherical harmonics is

differential_equations_c4b0ab745f013b338eb7833776770497aa78b408.png

Combining everything

Finally, combining the radial solution and the spherical harmonics into the final expression for our solutions:

differential_equations_d283bc856d7bfaaac98f85b4c355c5fa7411dcf2.png

where we have set differential_equations_9ef0b1f0d5ef41507550102e870660da21563257.png to ensure regularity at differential_equations_08836347d69b2e1f3f960129a55da2f494bbbdd0.png, and not have the following occur:

differential_equations_2a669f4ee8a99fd7939a742c712f50e62c05a2dc.png

Boundary conditions

  1. Establish that differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png must be periodic (since we're working in spherical coordinates) whose period evenly divides differential_equations_40ce122da09675999df03f3fff4b24d3df1132fb.png, which implies differential_equations_f56b556b5aef28b769c52c4c0bf2d3213e67eb2e.png and differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png is a linear combination of differential_equations_2a791df7f15faf124ba11955dab70dd425d69e4f.png
  2. Using what we observed in 1. and noting that differential_equations_80da8f06ee5dd9a03e6be24f145471d2fa9fffab.png is "regular" at the poles of the sphere, where differential_equations_0d62324a3f0c2ca80233459330db1c98d00ba8bc.png, we have a boundary condition for differential_equations_07aa11d29578308bc0642bd6f5d3e959f8d85582.png (which is a Sturm-Liouville problem) forcing the parameter differential_equations_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png to be of the form differential_equations_52fdd77e2e1fe4a635cb651410ed3eeb5cbc7335.png
  3. Making the change of variables differential_equations_5c6a9c87cfe8efb499a4c6b1cf166f18d5888ff3.png trnasforms this equation into the general Legendre equation, whose solution is a multiple of the associated Legendre Polynomial.
  4. Finally, the equation for differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png has solutions of the form

    differential_equations_02809fe8160cfeada18a4019eb638f033a2c67d9.png

  5. If we want the solution to be "regular" throughout differential_equations_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png, we have differential_equations_9ef0b1f0d5ef41507550102e870660da21563257.png.

When we say regular, we mean that the solution differential_equations_52755f5d394fd78c71203507d9439409a0d961b1.png has the property

differential_equations_7b345ea07061059c12da63e41772c201eb65c766.png

Proper description

A priori differential_equations_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is a complex number, BUT because differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png must be a periodic function whose period evenly divides differential_equations_40ce122da09675999df03f3fff4b24d3df1132fb.png, differential_equations_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is necessarily an integer and differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png is a linear combination of the complex exponentials differential_equations_2a791df7f15faf124ba11955dab70dd425d69e4f.png.

Further, the solution function differential_equations_80da8f06ee5dd9a03e6be24f145471d2fa9fffab.png is regular at the poles of the sphere, where differential_equations_0d62324a3f0c2ca80233459330db1c98d00ba8bc.png. Imposing this regularity in the solution differential_equations_07aa11d29578308bc0642bd6f5d3e959f8d85582.png of the second equation at the boundary of the domain is a Sturm-Liouville problem that forces the parameter differential_equations_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png to be of the form

differential_equations_d8c7bc5d3b57253d2cf59cee40d5d26b69ec54c5.png

Furthermore, a change of variables differential_equations_5c6a9c87cfe8efb499a4c6b1cf166f18d5888ff3.png transforms this equation into the (general) Legendre equation, whose solution is a multiple of the associated Legendre polynomial differential_equations_eda4e3006a3513107b82158dc054efb10f00b02a.png.

Finally, the equation for differential_equations_3d136f0fc4860468633907421c098b9feb0eef24.png has solutions of the form

differential_equations_fe39ac14c9ed43859844f830040cfbb1b689f4f1.png

And if we require differential_equations_2475f047fcda96d98d2d69c2383c13cb8823b2f0.png solution to be regular throughout differential_equations_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png forces differential_equations_9ef0b1f0d5ef41507550102e870660da21563257.png.

Solutions

Here, the solution assumed to have the special form differential_equations_fbfa6649835e25563dc0fd495c3397dc7d9086c4.png. For a given value of differential_equations_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png, there are differential_equations_0a422f8c3a9b20f6dcf198aa6ba6f6ec85f10ad8.png independent solutions of this form, one for each integer differential_equations_6ccf244556b9b20093fbc0a7f8acab7f2b7845b4.png. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

differential_equations_9df66e707a31a7ae8567bc62995e64dc4c1164dd.png

which fulfill

differential_equations_0f2e531756dbc917b9f2b11778f4c3fea48ca370.png

The general solution to the Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor differential_equations_46e0775b6eb543f4362375082ece2f890f157904.png,

differential_equations_aa953cfc06dd07a571b9963aedd423cfe4542ddb.png

where the differential_equations_6e0a3664ba119ac58b9f9d1027c8aff0053514f4.png are constants the factors differential_equations_39f0592a3498ed8c4e83cd4ad8fd3a86069d360d.png are known as solid harmonics.

Q & A

DONE Why do we actually require the differential_equations_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png to be an integer?

In solving for differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png we observe that

differential_equations_64ca2bebe1d756c0a9359c55c9572ed8aa97e8c3.png

Thus, if we require differential_equations_1cbaab2761a681fda45324e6ba33677e5130553e.png to be true, then we need differential_equations_05594e8ce8f52728892c4c673d8bc28149b4b24d.png to "go full circle" every differential_equations_9e283f0a770977b9c5776238e85e4b1f2406cdd4.png, hence differential_equations_f56b556b5aef28b769c52c4c0bf2d3213e67eb2e.png.

If it does NOT go full circle, then it would be discontinuous.

DONE We have TWO explanations for why we need differential_equations_52fdd77e2e1fe4a635cb651410ed3eeb5cbc7335.png, which one is true?
  • Question

    Here we state that differential_equations_c0e7e486574eda3063c12f9f7e267aa761dc1ee4.png has to be "because it's a Sturm-Liouville problem". I don't understand why it being a Sturm-Liouville problem necessarily imposes this restriction.

    Ah! Maybe it's because Sturm-Liouville problems require the limiting behavior of the boundary conditions to satisfy Lagrange's identity being zero!

    While here, we simply solve the differential equation involving differential_equations_2475f047fcda96d98d2d69c2383c13cb8823b2f0.png, and differential_equations_c0e7e486574eda3063c12f9f7e267aa761dc1ee4.png just falls out from this solution, and as far as I can tell, we're not actually saying anything about the boundary conditions in this case.

  • Answer

    From what I can understand, the eigenvalue differential_equations_c0e7e486574eda3063c12f9f7e267aa761dc1ee4.png simply falls out from differential_equations_2475f047fcda96d98d2d69c2383c13cb8823b2f0.png as we found, and differential_equations_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png being of this form does not seem to be related in any way to boundary conditions.

    What is related to the boundary conditions, the fact that differential_equations_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png is an integer.

    What is related to the b

Bessel equation

differential_equations_35140bf1a0c580fb20fc178b9286af67a54d802f.png

This can be solved using Frobenius method.

In the case where differential_equations_c8ef00da62fde5eacdcbe7585f781e43caa12229.png, we end up with the Bessel function of the 1st kind :

differential_equations_386c50b2631fe34607728c0589d1da6cf5e5d607.png

And the Bessel function of the 2nd kind :

differential_equations_a5302595e41beb7760c2c56029a6104fcd42ada0.png

Thus, the general solution will be the linear combination:

differential_equations_e3f4c11dbb3396a1c8ac94fa3ea57363308b8bec.png

Where we have the following limiting behavior:

differential_equations_f7a75a9e53957475e94860416614d3a4bbae7e86.png

Note: Quite often we want the solution to be "regular" / nicely behaved as differential_equations_6017c1de4abd9f51339bbc4adfe9d16fde77c778.png, thus we often end up setting differential_equations_c81a0d6869224500d74409893e63baa307c5a764.png in the expression for differential_equations_132744d49f7041ade4a8b3d79d895ee75ae3ab7e.png.

Problems

10.2 Fourier Series

Ex. 10.2.1

Assume that there is a Fourier series converging to the function differential_equations_93369077affb352dbdb97e8b3182fd50784f2b14.png defined by

differential_equations_23b404defd8d6b2bc2a5d48019eae176c6ab3952.png

with the property that

differential_equations_b61171a054ad8eac160a08d6fb2284b6c1344a5b.png

i.e. it's periodic with a period of 4.

Find the components of the Fourier series.

Solution

differential_equations_63dc56b31863ba2a95a425602a2be5b152b4132b.png

See p. 601 in (Boyce, 2013) for the full solution.

10.5 Separation of variables; Heat conduction in a Rod

Find the solution of the heat conduction problem

differential_equations_b08e544f7982b71cd0e6bf955176f979199535c6.png

Solution

differential_equations_fdf93f59ae3c6d4f6b00a6f5c64b4a65872aef82.png

We suppose

differential_equations_db3b4575ff34a930eb22b1fce60ea83330d6b009.png

And the boundary conditions are satisfied by

differential_equations_a68a9a954d51f1385a452d9083a8e0f9408caba6.png

First we note that with differential_equations_42683648b06176ce7644a8b4be60ac09f9660b5c.png above we get the following

differential_equations_fac79c7b3079f631aa55b938385a7f0e8fa38c2c.png

which gives

differential_equations_328e2934be89835936408a0932d633b82ad92b94.png

for some constant differential_equations_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, which gives us the system

differential_equations_f189ffb8f08a16d874ec5cf66c54a37b4bd0bbb5.png

The solution to the first equation simply gives us

differential_equations_903bc79f0783b5e2394b8f53e050385b03d4647d.png

Considering the boundary conditions, we get

differential_equations_e433efb73a82daa403201d221cfe4daed3a1967d.png

thus we get the eigenfunctions and eigenvalues for differential_equations_d675a6f657a0ed51e0ae3423cc858cc4f978e83c.png:

differential_equations_e9702b27e2aeb0b1e898b135e93d67b2d6e87e7a.png

Substituting these eigenvalues into the differential equation with differential_equations_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png:

differential_equations_c312efe03e108be31d84c767761b395819b5ddf3.png

which can be solved by use of integrating factors, thus

differential_equations_9faa95ff2c31d616805c4aeab5ff9e3688242946.png

Finally, combining both into the eigenfunctions for the entire solution differential_equations_dde23674dfca8d5e499e9ecf854bd00741657ddd.png

differential_equations_ce9a7caac1bde440e5d8dcee4e608a82756417a7.png

Which gives us the general solution:

differential_equations_a89064c3703b7fcb1b3ad0247219bc8c037c366f.png

and solving for the coefficients using the initial conditions, i.e. when differential_equations_9b50c3c5095e3def3ea0a38afe0602245b66e068.png

differential_equations_befacc16f91ca0362d7f52502755986c13f9a440.png

Hence the final solution is

differential_equations_c50e4d5e093cbabc929ddf570ecce31d4a471a3c.png