I am looking for help with solving the following Goursat problem (this is what the paper I am reading from calls it). I have been attempting a numerical solution in matlab but I do not fully understand the (internal?) boundary conditions and how to include them in my solution. For simplicity sake, I have been assuming $ q(x)=0 $, and for the project that I'm working on it doesn't matter whether or not $ q(x)=0 $. I'm looking for advice or at least the direction to some literature where I can educate my self on the matter.
$$ k_{xx} - k_{ss} + q(x)k = 0,\ \ 0<x<s<l\\k(0,s)=0, \ \ \ \ k(x,x)=-\frac 12\int_0^xq(\xi)d\xi $$
Please and thank you, Chris
I suppose that $q(\zeta)$ is a given function. Thus, the boundary condition $k(x,x)$ can be computed on each point of the boundary line.
So, all is known for the numetical compution on the discretized triangular domain.