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

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.

Answer 1

[Pavel Osipov]

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.

[Tanya Sharma]

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]

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

Answer 2

[Pavel Osipov]

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.

[Tanya Sharma]

Thanks Pavel!

But this will give me only five eigenvalues. As I am solving the eigenvalue of a differential equation and it can have many eigenvalues.

I am attaching the linearized eigenvalue problem. Is there a way I can find the unknown eigenvalues?

Answer 3

[Pavel Osipov]

See attachment