Solution of Convolution Integral Equations by the Method of Differential Inversion

Abstract

In this paper, a method called “differential inversion” is presented for solving a class of integral equations (Fredholm integral equations of the first kind) of convolution form. Differential inversion (DI) is presented as an alternative to Fourier transform methods. The method expresses the unknown function as a series of successive derivatives of the known function. The known-unknown functions are evaluated at the same point. This new expansion is not to be confused with a Taylor expansion, although we show that every Taylor expansion is a differential inversion expansion. Differential inversion is shown to be related to Gauss’s multipole expansion of a field at large distances from its source. The fundamental mathematical objects in this theory are generalizations of distributions, which we term “generalized distributions” or “hyperdistributions.” An important feature of our method is that convolutions form a closed algebraic field in the space of hyperdistributions. The relation of this field to Mikusiński’s field is discussed. A major result of our analysis is to give an explicit construction of the convolution inverse for any element of our field. The coefficients in the differential inversion series may be computed from the moments of the kernel. The inversion of convolution equations with exponential and Gaussian kernels is demonstrated in this paper (although the method may be applied to any kernel with finite moments). In particular, unique solutions are obtained in the space of hyperdistributions, even in those cases that cannot be represented by convergent Fourier integrals. The choices of kernels made here are relevant to a number of physical problems in signal processing, image processing, radiative transfer, and other inverse problems.