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I am trying to solve this problem on MATLAB of which code i have shared. It involves solving eigen values of a symbolic matrix with one variable 'f' and then using those values further in calculation. It is taking whole day what should i do?

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ABHISHEK KALYANWAT
ABHISHEK KALYANWAT on 5 Feb 2020
Edited: ABHISHEK KALYANWAT on 17 Feb 2020
syms f
l=0.1008;
T=0.005;
Dh=0.01;
D1=0.04;
D2=0.16;
x=20*(T*pi*Dh^2/4)/(0.1008*pi*D1^2/4);
c_0=619;
k=2*pi*f/c_0;
t=(0.006+1i*k*(T+0.75*Dh))/x;
Ka=(k^2)-((1i*4*k)/(D1*t));
Kb=(k^2)-((1i*4*k*D1)/((D2^2-D1^2)*t));
Delta=[-1 0 0 0; 0 -1 0 0; 0 0 Ka^2 Ka^2-k^2; 0 0 Kb^2-k^2 Kb^2]
g=eig(Delta);
ee=vpa(g,3)
C1=[1 1 1 1];
C2=[-((g(1,1)^2+Ka^2)/(Ka^2-k^2)) -((g(2,1)^2+Ka^2)/(Ka^2-k^2)) -((g(3,1)^2+Ka^2)/(Ka^2-k^2)) -((g(4,1)^2+Ka^2)/(Ka^2-k^2))];
C3=[1/g(1,1) 1/g(2,1) 1/g(3,1) 1/g(4,1)];
C4=[C2(1,1)/g(1,1) C2(1,2)/g(2,1) C2(1,3)/g(3,1) C2(1,4)/g(4,1)];
A1=[C3(1,1)*exp(g(1,1)*l) C3(1,2)*exp(g(2,1)*l) C3(1,3)*exp(g(3,1)*l) C3(1,4)*exp(g(4,1)*l)];
A2=[C4(1,1)*exp(g(1,1)*l) C4(1,2)*exp(g(2,1)*l) C4(1,3)*exp(g(3,1)*l) C4(1,4)*exp(g(4,1)*l)];
A3=[-exp(g(1,1)*l)/(i*k) -exp(g(2,1)*l)/(i*k) -exp(g(3,1)*l)/(i*k) -exp(g(4,1)*l)/(i*k)];
A4=[-(exp(g(1,1)*l)*C2(1,1))/(i*k) -(exp(g(1,1)*l)*C2(1,2))/(i*k) -(exp(g(1,1)*l)*C2(1,3))/(i*k) -(exp(g(1,1)*l)*C2(1,4))/(i*k)];
E1=[C3(1,1) C3(1,2) C3(1,3) C3(1,4)];
E2=[C4(1,1) C4(1,2) C4(1,3) C4(1,4)];
E3=[1/(i*k) 1/(i*k) 1/(i*k) 1/(i*k)];
E4=[-C2(1,1)/(i*k) -C2(1,2)/(i*k) -C2(1,3)/(i*k) -C2(1,4)/(i*k)]
A=[A1; A2; A3; A4]
I=eye(4);
invA=A\I;
E=[E1; E2; E3; E4];
PP=E*invA;
vpa(PP,3)

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Walter Roberson
Walter Roberson on 16 Feb 2020
where D=d/dz (differential operator)
No, if it were the differential operator then D could not appear by itself in the Δmatrix: it would have to be followed by an expression or variable. It would also be quite unlikely to be followed by or similar: if you were taking the derivative of then you would skip the + in the expression.
It looks to me as if D is a variable that is being used linearly in entries in row 3 and 4.
Possibly D is the derivative of something particular, but if so then we would have to know what it is the derivative of.
That equation does not look to me like something you would solve with eigenvalues: that is an equation for simultaneous equations. With the right hand side being all 0, you should probably be thinking about null spaces rather than eigenvalues.
David Goodmanson
David Goodmanson on 16 Feb 2020
Hi Abhishek
could you post the entire set of equations for this problem, including the definition of D? It appears that D could be d/dz, since after doing the mattix multiplication delta*y, D would be operating on the components of y.
ABHISHEK KALYANWAT
ABHISHEK KALYANWAT on 17 Feb 2020
Thank you so much for your comments.
Here is the background of 'Delta Matrix' here notation used for this matrix is :
w=2*pi*f (f is frequency) , Everything else is constant. M1 ,M2 are 0 for this particular case.
are the zeros of the determinant of the coefficient matrix of equation (6a).
I have used 'PP' notation for matrix .
PREVIOUSLY I WAS TRYING TO CALCULATE EIGEN VECTORS TO FIND ZEROS OF THE DETERMINANT OF THE COEFFICIENT MATRIX WHICH IS TOTALLY INCORRECT. EXCUSE ME FOR MY LIMITED UNDERSTANDING OF MATHEMATICS.
Delta=[-1 0 D 0; 0 -1 0 D; D 0 Ka^2 Ka^2-k^2; 0 D Kb^2-k^2 Kb^2]
CAN YOU TELL ME HOW CODE SHOULD BE WRITTEN TO FIND ZEROS OF THE DETERMINANT OF DELTA MATRIX.
THANK YOU.

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ABHISHEK KALYANWAT
ABHISHEK KALYANWAT on 11 Feb 2020
Thank you so much David ,walter and Darova for your answers. Trying to solve it numerically seems worth trying here.

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