Fredholm Integro-Differential Equations

Abstract

In Chapter 2, the conversion of boundary value problems to Fredholm integral equations was presented. However, the research work in this field resulted in a new specific topic, where both differential and integral operators appeared together in the same equation. This new type of equations, with constant limits of integration, was termed as Fredholm integro-differential equations, given in the form

$${u^{\left( n \right)}}\left( x \right) = f\left( x \right) + \int_a^b {K\left( {x,t} \right)u\left( t \right)dt,{u^{\left( k \right)}}\left( 0 \right) = {b_k},0 \leqslant k \leqslant n - 1,} $$

(6.1)

where ${u^{\left( n \right)}}\left( x \right) = \frac{{{d^n}u}}{{d{x^n}}}$. Because the resulted equation in (6.1) combines the differential operator and the integral operator, then it is necessary to define initial conditions u(0), u′ (0), , u ⁽ⁿ⁻¹⁾(0) for the determination of the particular solution u(x) of equation (6.1). Any Fredholm integro-differential equation is characterized by the existence of one or more of the derivatives u′, (x), u″(x), outside the integral sign. The Fredholm integro-differential equations of the second kind appear in a variety of scientific applications such as the theory of signal processing and neural networks [1–3].