I'm looking for any insight into solving the following PDE:

$$u_{tt}=c^2 (u_{xx}+u_{yy})-\sin(y)$$ $$u=0, y\in {0,\pi} $$ $$u_x=0, x\in {0,1}$$ $$u(x,y,0)=\cos(\pi x)\sin(3y) $$ $$u_t(x,y,0)=0$$

In particular, if I was to attempt a Fourier series solution to the problem, given that both the equation and initial conditions are inhomogeneous, what is the thought process of guessing a solution form? I tried guessing a form like $$ \sum\limits_{i=1}^\infty A_n(y,t)\cos(\pi nx) $$ then I tried to find the coefficients $A_n$ by matching the initial conditions first, and then the boundary conditions, but I couldn't get this to lead anywhere. Any help would be much appreciated!

  • $\begingroup$ Look by a solution with the form $A(t)sin(y)+B(t)cos(\pi x)sin(3y)$ $\endgroup$ – Juan Ospina Dec 11 '14 at 18:36

The solution is

$u \left( x,y,t \right) =\cos \left( \pi \,x \right) \sin \left( 3\,y \right) \cos \left( c\sqrt {{\pi }^{2}+9}t \right) +{\frac {\sin \left( y \right) \cos \left( ct \right) }{{c}^{2}}}-{\frac {\sin \left( y \right) }{{c}^{2}}} $

  • $\begingroup$ Thank you very much, Juan. So I can see my initial guess at a general form for the solution is wrong, what was your thought process to decide the most general form of the solution (before solving for the coefficients)? $\endgroup$ – Z W Dec 11 '14 at 3:13

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