A Study on Integral Equation and Its Applications

Abstract

The present survey paper samples recent advances in the numerical analysis of Volterra integral equations of the first and second kind and of integro-differential equations (including equations with weakly singular kernels); except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years. A fairly extensive bibliography (selected to be representative rather than comprehensive) complements the paper.

The study also presents the Integral transform, mathematical operator that produces a new function f(y) by integrating the product of an existing function F(x) and a so-called kernel function K(x, y) between suitable limits. The process, which is called transformation, is symbolized by the equation f(y) = ∫K(x, y)F(x)dx. Several transforms are commonly named for the mathematicians who introduced them: in the Laplace transform, the kernel is e−xy and the limits of integration are zero and plus infinity; in the Fourier transform, the kernel is (2π)−1/2e−ixy and the limits are minus and plus infinity. The study analysis the volterra integral equation and kernel types and presents the examples solution for the discussed problems.