2
$\begingroup$

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.

$\endgroup$
7
  • 1
    $\begingroup$ 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 $f$. $\endgroup$
    – user14717
    Feb 23, 2017 at 4:17
  • 1
    $\begingroup$ 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. $\endgroup$
    – user14717
    Feb 24, 2017 at 19:22
  • $\begingroup$ @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. $\endgroup$
    – Giuseppe Negro
    Feb 27, 2017 at 17:51
  • $\begingroup$ I have the first (2001) edition, ISBN 1-58488-110-0. $\endgroup$
    – user14717
    Feb 27, 2017 at 18:13
  • $\begingroup$ 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. $\endgroup$
    – user14717
    Feb 27, 2017 at 22:28

0

Reset to default

You must log in to answer this question.