How to find eigenvalues for a system of lenearized ordinary differential equations?

Question

Tanya Sharma 2019-10-5

2
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/483702-how-to-find-eigenvalues-for-a-system-of-lenearized-ordinary-differential-equations

评论： Poly 2022-3-10

采纳的回答： Pavel Osipov

I have a system of linearized ODEs with corresponding boundary conditions.

%----------------------------system of ODEs--------------------------------------%

y'(1)=y(2)

y'(2)=y(3)

y'(3)=(phi./Da).*y(2)+(2.*phi.*Fr./A1).*fd.*y(2)-(fd1.*1./A1).*y(3)-(fdd.*1./A1).*y(1)+(2.*fd.*1./A1).*y(2)-(e./A1).*y(2)-(phi.*Ra./(A1^2).*A2).*y(4)

y'(4)=y(5)

y'(5)=-(Pr./A2).*(fd.*y(5)+thd.*y(1)+e.*y(4))];

%---------------------------boundary conditions----------------------------------%

y(1)=y(2)=y(4)=0 at eta=0

y(2)=y(4)=0 at eta=0;

here Pr phi Ra Da Fr A1 A2 fd1 fd fdd thd are known quantities and 'e' is unknown.

I need to solve the system to find out the eigenvalues (e).

Thanks in advance.

0 个评论
显示 -2更早的评论隐藏 -2更早的评论

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

Pavel Osipov 2019-10-5

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/483702-how-to-find-eigenvalues-for-a-system-of-lenearized-ordinary-differential-equations#answer_394932

Tanya, hi.

write so:

dyi/dt =...y1 (t)+...y2 (t)+...+y5(t);

let x (t)=[y1;y2;...;y5]; ->

((V/ve) x=Ah; A - matrix coeff. Your system. Let's formally denote d/dt=p

px-Ax=0; - > (p*E-A) x=0; since x is not 0, then

det(p*E-A)=0. This is the equation for the eigenvalues of p.

2 个评论
显示无隐藏无

Tanya Sharma 2019-10-9

The unkown 'e' is already in the equations y'(3) and y'(4). I want to find all the possible eigenvalues 'e' for this problem. Can you explain again?

Thanks in advance.

Poly 2022-3-10

Hello Tanya!! I don't know whether you get the code right or not? but can you share the code if possible

请先登录，再进行评论。

Answer 2

Pavel Osipov 2019-10-9

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/483702-how-to-find-eigenvalues-for-a-system-of-lenearized-ordinary-differential-equations#answer_395472

Hi!

det(p*E-A)=0. This is the equation for the eigenvalues of p. - -> The unkown "p" is solution det(p*E-A)=0. det - is the determinant with dimensions 5x5.

px-Ax=0 ->Ax=px, p is eigenvalues of A MATLAB command [V,D] = eig(A) returns diagonal matrix D of eigenvalues and matrix V whose columns are the corresponding right eigenvectors, so that A*V = V*D. (from MATLAB help).

eigenvalues p is 5x1 vector = liagonal elements D. eigenvectors of A see at columns V.