Initial-Boundary-Value Problem for an Integrodifferential Equation of the Third Order

Abstract

We study the initial-boundary-value problem for an integrodifferential equation of the third order. Replacing the required function and its time derivative by a combination of two new unknown functions, we reduce this problem to an equivalent problem in the form of a family of multipoint problems for a system of two Volterra integrodifferential equations of the first order and integral relations. We construct algorithms for finding the solution of the equivalent problem. By the method of parametrization, we establish the conditions for the unique solvability of a family of multipoint problems for the system of first-order Volterra integrodifferential equations in terms of the initial data. We also establish the conditions for the existence of the unique classical solution of the initial-boundary-value problem for the integrodifferential equation of the third order in terms of the coefficients of this equation and boundary functions.