Solving Diffuse Light Simulation with Double Integral Equation?

[Luonnos]

Recently, I've been working on a program to simulate diffuse light, and I've hit a snag. I need to solve (at least so that a computer can compute L(x) quickly) something of the form:

$$L(x)=T(x)+c\int_0^{l_2}\int_0^{l_1} W(x,u_1,u_2) L(u_1) du_1 du_2$$

W and T are pretty well behaved, and I can compute them and their integrals in a small amount of time. I'm pretty new to differential and integral equations, so any help would be appreciated.

 

[phyzguy]

If you discretize this on a computer, then L will be a column vector with dimension equal to the number of grid points. First, you should be able to integrate out u2, since L is a function only of u1. Then I think you can write your equation as follows, where L and T are column vectors, c is a constant, I is the identity matrix, and W is a square matrix:
$$L = T + c W L$$
$$(I-c W) L = T$$
$$L = (I-c W)^{-1} T$$
Then you can use a linear algebra solver like BLAS or LAPACK to solve for L. These will find solutions even if the matrix dimensions are on the order of 1000's x 1000's.