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

Asked by Tanya Sharma on 5 Oct 2019
Latest activity Answered by Pavel Osipov on 24 Nov 2019 at 21:39
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.

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3 Answers

Answer by Pavel Osipov on 5 Oct 2019
 Accepted Answer

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.

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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.

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Answer by Pavel Osipov on 9 Oct 2019

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.

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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?

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Answer by Pavel Osipov on 24 Nov 2019 at 21:39

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