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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.

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    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 $f$. – 2017-02-23
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    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. – 2017-02-24
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    @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. – 2017-02-27
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    I have the first (2001) edition, ISBN 1-58488-110-0. – 2017-02-27
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    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. – 2017-02-27
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    @user14717: Assuming $t\ge 0 $ would not be too much of a problem. I have checked your reference, and I agree that one can obtain a Green's function for this problem. It is going to be given by an infinite sum involving Bessel's functions, however. – 2017-02-27
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    What is your software stack? If you are using C++, boost::math will have very fast algorithms for the evaluation of Bessel functions. I agree that a nice FD algorithm might be preferable but coding time will be longer . . . – 2017-02-27

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