Numerical solution for Fredholm first kind integral equations occurring in synthesis of electromagnetic fields

It is known that Fredholm integral equations of the first kind integral(1)(0) k(s,t)x(t)dt = y(s), s is an element of[0,1] (1) \with the kernel (s - t)(m) /[1-(s-t)(2)](n) occur when solving with problems of synthesis of electrostatic and magnetic fields (m, n - nonnegative rational numbers). This paper presents two approaches for solving such an equation. The first one involves discretization by a collocation method and numerical solution using an approximate orthogonalization algorithm. The second method is based on a nature inspired heuristic, namely genetic programming. It applies genetically-inspired operators to populations of potential solutions in the form of program trees, in an iterative fashion, creating new populations while searching for an optimal or near-optimal solution to the problem at hand. Results obtained in experiments are presented for both approaches.