# Numerical methods to compute radially symmetric solutions to the wave equation in high dimension

I have the necessity to compute approximate solutions to the equation (pedices denote derivatives) $$\tag{W} u_{tt}-u_{rr}-\frac{n-1}{r}u_r=f(t, r)\quad t\in\mathbb R,\ r\in [0, R]$$ subject to the initial conditions $$u(0, r)=0,\quad u_t(0, r)=0$$ and the boundary conditions $$u_r(t, 0)=u(t, R)=0.$$ The equation (W) is obtained by writing the wave equation $u_{tt}-\Delta u = f$ in spherical polar coordinates in $\mathbb R^n$, under the assumption that the source term $f$ is radially symmetric at all times. I am especially interested in the value $n=5$ for the spatial dimension.

Can you give me to some numerical scheme to compute such a solution numerically?

Unfortunately, I am not familiar with numerical analysis. I would be glad to receive an answer or literature pointer containing some details on how to implement such a method.

Thank you for reading.

• Given the simplicity of the equation, I would probably look for a Green's function and then do a numerical integration. Since you're only in 2d, you should be able to compute this integral super fast assuming some regularity on $f$. Feb 23 '17 at 4:17
• The Green's function for this equation in $n=3$ is given in Duffy's Green's functions with applications, Chapter 3, exercise 8. To get it to $n=5$ will require changing the source to $\delta(r-r')/(A(\mathcal{S}^{4})r^{4})$. Then you'll need to use some series acceleration to make it viable. Feb 24 '17 at 19:22
• @user14717: The method you suggest and the reference you gave are interesting, but unfortunately, I cannot find the result you mention. Which edition of the book are you referring to? I have consulted 2nd edition, CRC press, 2015. Feb 27 '17 at 17:51
• I have the first (2001) edition, ISBN 1-58488-110-0. Feb 27 '17 at 18:13
• Do you mean for $t \in \mathbb{R}_{\ge 0}$ so that the signal is causal? If so, this will allow use of the Laplace transform in generating the Green's function-way simpler. Feb 27 '17 at 22:28