清空筛选条件
清空筛选条件

How can numerically compute eigenvalues of an ordinary differential equation in MATLAB?

25 次查看（过去 30 天）
Lemuel Carlos Ramos Arzola
评论： Lemuel Carlos Ramos Arzola 2019-2-15
Hello,
I need to compute (numerically) the eigenvalues (L) of this singular ODE,
, subject to
Is it possible to use the Matlab function bvp4c? Or another?
Best regards,
Lemuel

请先登录，再回答此问题。

采纳的回答

Lemuel Carlos Ramos Arzola
Hello,
Thanks for the answers.
I found the same problem here:
https://www.mathworks.com/matlabcentral/answers/405977-singular-jacobian-with-bvp4c-used-to-solve-eigenvalue-problem
and the answer completely satisfies me
  4 个评论
显示 2更早的评论
Torsten
Torsten 2019-2-14
But as far as I see, you won't get an eigenvalue for an arbitrary choice of the third boundary condition.
E.g. if you have the ODE
y''+L*y = 0
y(0)=y(2*pi)=0,
the eigenvalues and eigenfunctions are L_n = (n/2)^2 and y_n(x) = sin(n*x/2) (n=1,2,3,...).
So if you choose y'(0)=1 as third boundary condition at x=0, e.g., every function y(x)=a*sin(sqrt(L)*x) with a*sqrt(L)=1 is a solution of the ODE, not only those for which a=2/n and L=(n/2)^2 (n=1,2,3.,,,).
Lemuel Carlos Ramos Arzola
Lemuel Carlos Ramos Arzola 2019-2-15
Dear Torsten,
Thanks for all your comments.
The singular ODE (the original in this question) arises in the context of heat convection in tubes. This problem is know as Graetz's problem.
Recently I "found" that a third boundary condition exist, namely, y(0)=1. With this "new" condition, the MATLAB function bvp4c can also find unknown parameters in the ODE, in our case, L.
I only need to change the initial guess for the unknown parameter L, not the third boundary condition as I said wrongly in the previous comment .
Following the algorithm shown in this link, and changing the initial guess for L, I found the eigenvalues that I needed.
https://www.mathworks.com/matlabcentral/answers/405977-singular-jacobian-with-bvp4c-used-to-solve-eigenvalue-problem
Although I had some difficulties with the singularity at x = 0.

请先登录，再进行评论。

更多回答（2 个）

Bjorn Gustavsson
Bjorn Gustavsson 2019-2-11
Have a look at what you can do with chebfun. It seem to cover eigenvalue/eigenfunctions of ODEs in some detail:
Chebfun ODE-eigenvalue examples
HTH
Torsten
Torsten 2019-2-11
https://www.wolframalpha.com/input/?i=y''(x)%2B1%2Fx*y'(x)%2Ba*(1-x%5E2)*y(x)%3D0,+y'(0)%3D0
So you are left with the problem to find "a" such that
L_(0.25*(sqrt(a)-2)) (x) = 0 for x=sqrt(a).

类别

在 Help Center 和 File Exchange 中查找有关 Ordinary Differential Equations 的更多信息

标签

产品


版本

R2018a

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by