# Solving Time-Dependent PDE's Analytically with no non-trivial Boundary Conditions

I'm having some trouble solving a partial differential equation that is supposed to model the propagation of a scalar field in Schwarzschild spacetime using spherical coordinates. I'm using the DSolve function, but Mathematica keeps on returning my original differential equation when I try to apply DSolve. The only boundary condition that I can think of without loss of generality in the problem is when t = 0, . Is there any way I can get an analytical solution to this time dependent PDE?

    eqn  =
((r - 2 M)/r)*
D[u[t, r, \[Theta], \[Phi]], {t,
2}] == ((2*r^2 - 6*r*M)/(r - 2 M)^2)*
D[u[t, r, \[Theta], \[Phi]], {r,
1}] + (r^3/(r - 2 M)) *
D[ u[t, r, \[Theta], \[Phi]], {r, 2}] +
(r^2 * Cot[\[Theta]])*
D[u[t, r, \[Theta], \[Phi]], {\[Theta], 1}] +
(r^2) * D[u[t, r, \[Theta], \[Phi]], {\[Theta], 2}] +
(r^2 * Sin[\[Theta]]) *
D[u[t, r, \[Theta], \[Phi]], {\[Phi], 2}]

DSolve[{eqn, u[0, r, \[Theta], \[Phi]] ==
0} , u[t,
r, \[Theta], \[Phi]], {t, r, \[Theta], \[Phi]}]

• most probably not. DSolve cannot give you an analytical solution for all equations, only the ones that knows how to solve. Your problem has nothing to do with the boundary conditions I would say. Try to go numerical. – tsuresuregusa Apr 29 '16 at 15:03
• Because eqn is linear, the solution with your boundary condition is u == 0. – bbgodfrey May 1 '16 at 1:34