In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schrodinger equation for the Morse potential that has an analytically...

Under the assumption that A is the generator of a twice integrated cosine family and K is a scalar valued kernel, we solve the singular perturbation problem (E2) 2 2u′′ 2 (t) + u ′ 2(t) = Au2(t) + (K ∗Au2)(t) + f2(t), (t ≥ 0)(2 > 0), when 2 → 0, for the integrodifferential equation (E) w′(t) = Aw(t) + (K ∗Aw)(t) + f(t), (t ≥ 0), on a Banach space. If the kernel K verifies some regularity condit...

We study the singular perturbation problem (E2) 2 2u′′ 2 (t) + u ′ 2(t) = Au2(t) + (K ∗Au2)(t) + f2(t), t ≥ 0, 2 > 0, for the integrodifferential equation (E) w′(t) = Aw(t) + (K ∗Aw)(t) + f(t), t ≥ 0, in a Banach space, when 2 → 0. Under the assumption that A is the generator of a strongly continuous cosine family and under some regularity conditions on the scalar-valued kernel K we show that p...

Waves in chemically excitable systems can refract when impinging on an interface between regions of different reaction kinetics and/or diffusion constants. Here we study this process using the thin reaction-zone limit wherein the dynamics of the system can be reduced to the tracking of the boundaries between quiescent and excited regions. We show how to derive an integrodifferential equation fo...

In this paper we study the numerical solution of nonlinear Volterra integrodifferential equations with infinite delay by spline collocation and related Runge-Kutta type methods. The kernel function in these equations is of the form k(t,s,y(t),y(s)), with a representative example given by Volterra's population equation, where we have k(t, s, y(t),y(s)) = a(t s) ■ G(y(t), y(s)). '

A strong limit theorem is proved for a version of the well-known Kac-Zwanzig model, in which a “distinguished” particle is coupled to a bath of N free particles through linear springs with random stiffness. It is shown that the evolution of the distinguished particle, albeit generated from a deterministic set of dynamical equations, converges pathwise toward the solution of an integrodifferenti...

and Applied Analysis 3 We note that 1.1 in its general form involves some different types of differential and difference equations depending on the choice of the time scale T . For example: 1 for T R, we have σ t t, μ t 0, and xΔ t x′ t , and 1.1 becomes the Cauchy integrodifferential equation: x′ t f ( t, x t , ∫ t 0 k t, s, x s ds ) , t ∈ R,