Novel computational approach to solve convolutional integral equations: method of sampling for one dimension [Online First]

Abstract

Objective: This paper proposes a new methodology to solve one-dimensional cases of integral equations with difference kernels using Fourier analysis. Methodology: In this study, it was proven that any Fredholm equation of the first kind can be expressed as an extended convolutional problem; consequently, a new approach to solve that problem, using the nonideal instantaneous sampling theory and Fourier analysis, can be developed. Results and Discussion: The proposal was extensively evaluated and compared with the method of moments by considering two benchmarks. The first was a narrowband problem related to a second-order differential equation with specific boundaries. The second was a standard wideband problem related to wire antenna radiation in electrodynamics, known as the Pocklington equation. In both cases, we derived new interpretations and different approaches to solve the problems efficiently. Conclusions: The new proposal generalized the method of moments via new interpretations, strategies and design rules. We found that the techniques based on the method of moments are point-matching procedures independent of the weighting functions; the basis functions can be designed as generalized interpolation functions with more information provided by the original domain; the weighting functions literally represent a sampled linear filter; the unknown continuous function can be approximated without using the classical variational approach; and several new strategies based on the Fourier transform can be used to reduce the computational cost