A Numerical Method for Solving Fredholm Integral Equations of the First Kind Using Singular Values

Abstract

The integral equation in question is approximated by simple numerical quadrature formulas plus collocation.Each row of the resulting matrix equation for the unknown function values is weighted by the reciprocal of the standard deviation of the known function.A singular value decomposition is used to obtain a solution for the resulting linear system. By avoiding the use of the smallest singular values, an approximate solution is calculated which frequently solves the problem “close enough” and removes a great deal of the oscillations in the solution which are inherently present due to the ill-posed nature of the problem.Test cases and computational results are presented.