Partial differential equation reduced but

[derdack]

Hi everybody... I have problem with this system of equation

d^4 (T1[t])/dt^4 + (A1 - B1*f[x])*d^2 (T1[t])/dt^2 + (C1 + D1*f[x])*
T1[t] - E1*T2[t] = 0

d^4 (T2[t])/dt^4 + (A2 - B2*f[x])*d^2 (T2[t])/dt^2 + (C2 + D2*f[x])*
T2[t] - E2*T1[t] = 0

X[x] = (Ch[k*x] + Cos[k*x])/(Ch[k*l] - Cos[k*l]) - (Sh[k*x] +
Sin[k*x])/(Sh[k*l] - Sin[k*l]);

f[x] = X''[x]/X[x];

A1,B1,C1,D1,E1,A2,B2,C2,D2,E2,l,l are constants. The solutions are in this form e^i*r*t. And than I can get a determinant which must be equal to zero because of non trivial solutions. Form is

r[x]^4-V1[x]*r[x]^2+P1[x] -S1
-S2 r[x]^4-V2[x]*r[x]^2+P2[x]

S1,S2 are constants

I need a solutions r, not a solutions r as a function of x. PLEASE! Ritz method can help but ...

Derdack in big problem...

 

[jvicens]

Hi Derdack,

Thank you for sharing your problem with us. It seems like you are working with a system of coupled differential equations, and you are trying to find the solutions in the form of exponential functions. Your goal is to find the values of r that satisfy the determinant equation for non-trivial solutions.

One approach you could take is to use the Ritz method, as you mentioned. This method involves approximating the solutions with a trial function that satisfies the boundary conditions of the problem. Then, you can plug this trial function into the differential equations and solve for the unknown coefficients. This method can be useful for problems with complicated boundary conditions, like yours.

Another approach you could try is to use a numerical method, such as the shooting method or finite difference method. These methods involve discretizing the differential equations and solving them numerically. The shooting method involves guessing initial values for the solutions and adjusting them until the boundary conditions are satisfied. The finite difference method involves approximating the derivatives with finite differences and solving the resulting system of equations.

I would also recommend checking your equations and boundary conditions to make sure they are correct. It may also be helpful to consult with a colleague or a professor for further guidance on how to approach this problem.

I wish you the best of luck in finding a solution to your problem. Don't give up, and keep exploring different methods and techniques. Science and research can be challenging, but the satisfaction of finding a solution is worth the effort. Good luck!