The following MATLAB script implements the formulas derived in
S. Sauter and A. Veit. Exact Solutions of Retarded Boundary Integral Equations. Preprint 03-2011, Universität Zürich,
which can be downloaded here.
The script comp_phi.m computes the solution of the retarded boundary integral equation SΦ=g on the unit sphere, where S is the time-domain single layer potential and the given right-hand side g is of the form g(x,t)=g(t)Ynm with n=0,1. Here Ynm denotes a spherical harmonic of degree n. In order to compute these solutions the first time derivative of g(t) must be known. A description how to use the MATLAB script is commented in the file. A script that computes the solution for general n will be available soon.
Example 1:
We consider the purely time-dependent right-hand side g(x,t)=t4e-2t. Then the following code in MATLAB
leads to the following solution of the boundary integral equation.
The animation below shows a visualization of this solution on the sphere.
Example 2:
We consider the right-hand side g(x,t)=t sin(2t) e-tY10. Then the following code in MATLAB
leads to the following solution of the one dimensional problem.
The animation below shows a visualization of the 3d solution phi*Y10 on the sphere.