Volterra-Fredholm Integral Equations

Abstract

The Volterra-Fredholm integral equations [1–2] arise from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and biological models. The Volterra-Fredholm integral equations appear in the literature in two forms, namely $$u\left( x \right) = f\left( x \right) = {\lambda _1}\int_0^x {{K_1}\left( {x,t} \right)u\left( t \right)dt + {\lambda _2}} \int_a^b {{K_2}\left( {x,t} \right)u\left( t \right)dt} ,$$ (8.1) and the mixed form $$u\left( x \right) = f\left( x \right) = \lambda \int_0^x {\int_a^b {K\left( {r,t} \right)u\left( t \right)dtdr} ,} $$ (8.2) where f(x) and K(x, t) are analytic functions. It is interesting to note that (8.1) contains disjoint Volterra and Fredholm integrals, whereas (8.2) contains mixed Volterra and Fredholm integrals. Moreover, the unknown functions u(x) appears inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of first kind. Examples of the two types of the Volterra-Fredholm integral equations of the second kind are given by $$u\left( x \right) = 6x + 3{x^2} + 2 - \int_0^x {xu\left( t \right)dt - \int_0^1 {tu\left( t \right)dt} ,} $$ (8.3) and $$u\left( x \right) = x + \frac{{17}}{2}{x^2} - \int_0^x {\int_0^1 {\left( {r - t} \right)u\left( t \right)drdt} .} $$ (8.4)