# Exact Solutions of Retarded Boundary Integral Equations

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,

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

g_prime=@(t) (4*t.^3-2*t.^4).*exp(-2*t);
t0=linspace(0,10,200);
phi=compute_phi(0,g_prime,t0);
plot(t0,phi);

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

g_prime=@(t) -sin(2*t).*exp(-t).* ...
(-4.*t.*cos(2*t)-sin(2*t)+sin(2*t).*t);
t0=linspace(0,10,200);
phi=compute_phi(1,g_prime,t0);
plot(t0,phi);

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.  