On the reduction of integro-differential equations
Griffith C. Evans · Transactions of the American Mathematical Society · 1914
Certain integro-differential equat ons can be completely treated by first resolving an integral equation, and second, integrating a differential equation.In so far, such equations do not constitute a new problem in analysis, because they are reducible to equations of simpler types.From this point of view Volterra discusses the equation V2V(x,y,z\t)+ f f(t,r)V2V(x,y,z\r)dr =F(x,y,z,t) Jtowhich is satisfied by the potential V of an electric field in an isotropic medium, in the so-called "static case" of hysteresis.tBy solving this equation, regarded as an integral equation, for V2V, we are led to the differential equation of Poisson to determine V.We may however have the case that an integro-differential equation whose solution is subject to certain boundary conditions is reducible to one or more problems in equations of more elementary character only when account is taken of these conditions.This is the point of view to be developed in the present paper.It is a point of view which can be extended to more general types of functional equations.We shall treat, in the following pages, the reduction of boundary value problems in the case of the integro-differential equation of parabolic type, and quite briefly that of hyperbolic type, to problems in linear differential and integral equations separately.It is worth while perhaps to specially mention § 2, in which the generalization of the integro-differential equation of the second order to an equation of the first order involving integration * Presented to the Society, December