A New Wavelet Packet Bases to Solve Fredholm’s Integral Equations of the First Kind

Abstract

Different high order discretization methods and numerical expansions have been developed to find approximate solutions to integral equations with different kernels. However, direct application of standard numerical methods to the matrices obtained by discretizations of these equations can produce meaningless solutions. If the kernel is continuous, smooth and bounded, the integral operator is compact.
In this context Fredholm’s Integral Equations of the First Kind, i.e, K f(x) = ∫ ba h(x, y)f(y)dy = g(x), where f is the solution function and g is the data, are in general ill-conditioned inverse problems.
However, restrictions to finite dimensional spaces where the unknown function f and g live can assure existence, unicity and well-conditioned of the problem. In this work we construct approximate solutions to inverse problems associated to integral operators of the first kind applying a wavelet decomposition.
We restrict the problem to a bounded set of frequencies and we approximate the eigenfunctions of the operator from the images of finite set of wavelets functions trough the operator. Based on some properties of the basis, the resulting scheme is numerically stable.