I used Galerkin's method to solve a coupled PDE'S, I assumed a solution of the form exp( λ), in order to convert the DE'S into polynomial of λ. one of the equations is not homogeneous and the term exp( λ) in the 2nd equation did not cancel out, can any body help me regarding this issue
I am not sure I have understood your problem correctly. However, if you want to find a solution for the inhomogeneous problem with the help of a decomposition of the solution in exponentials then I think you should decompose also the known source in exponentials. As a result, the PDEs should reduce to ODEs. The general solutuion is the sum of the solution of these ODEs and an arbitrary linear combination of solutions of the homogeneous problem. Finally, boundary and initial conditions will remove ambiguities.
If an equation is inhomogeneous, you should (1) first solve the homogeneous equations first and get the eigen-functions (Sturm-Liouville problem). When the coefficients of these homogeneous equations are constant, you can assume that the eigen-functions are in the form of exp(lamba*x) as you did. (2) the solution can now be written as a sum of eigen-functions with unknown coefficients (called modal expansion) and the inhomogeneous term should be expanded in terms of the same eigen-functions found in (1), as suggested by Andrea Di Vata. (3) the resultant equations are now a set of ODEs for the unknown coefficients, which are easier to solve.
I will use a single inhomogeneous PDE to demonstrate the method. The idea is also valid for coupled inhomogeneous PDEs. Please see the attached PDF file. You may come back to me if you cannot understand it.
Dear Sir
Thank you very much for your assistance. I started the solution as you suggested in your previous answer, but I am not sure I understand the second step . Can you please direct me to more detailed solution steps. I have attached the equations along with some work in the solution.
Is P a constant or a function of x? Is H a constant or a function of x? What about delta?
Is z_s a constant? Could H be Heaviside function and delta the Dirac delta function?
I am sorry I didnot mention that, P is and Z_s are constants, delta is the dirac delta function and H is the Heaviside function. All other symbols are constants eccept C(k) which is theodorson function ( function of the frequency) and E* ( also G*) is the dynamic modulus ( the material is viscoelastic)
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