An integral-boundary value problem for a partial differential equation of second order

Anar T Assanova

doi:10.3906/mat-1903-111

Abstract

An integral-boundary value problem for a hyperbolic partial differential equation in two independent variables is considered. By introducing additional functional parameters, we investigate the solvability of the problem and develop an algorithm for finding its approximate solutions. The problem is reduced to an equivalent one, consisting of the Goursat problem for a hyperbolic equation with parameters and boundary value problems with an integral condition for ODEs with respect to the parameters entered. We propose an algorithm to find an approximate solution to the original problem, which is based on the algorithm for finding a solution to the equivalent problem. The convergence of the algorithms is proved. A coefficient criterion for the unique solvability of the integral-boundary value problem is established.

Highlights

On the domain Ω = [0, T ] × [0, ω], we consider the integral-boundary value problem for the hyperbolic equation of second order ∂2u ∂u= A(t, x) + B(t, x) + C(t, x)u + f (t, x), (1) ∂t∂x ∂x ∂t ∫aP (t)u(t, 0) + K(t, ξ)u(t, ξ)dξ = ψ(t), t ∈ [0, T ], (2)
1) Assuming u(t, x) = 0, w(t, x) = 0, λ(x) = 0, on the right-hand side of equation (15), we find initial approximations μ (0)(t), μ(0)(t), t ∈ [0, T ], from the boundary value problem with integral condition (15) and (12)
Assuming u(t, x) = 0, v(t, x) = 0, μ(t) = 0 on the right-hand side of equation (16), we find initial approximations λ (0)(x), λ(0)(x), x ∈ [0, ω], from the boundary value problem with integral condition (16) and (11)

Summary

Introduction

Some types of integral-boundary value problems for hyperbolic equations were studied in [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] Solvability conditions for these problems are established in different terms. A boundary value problem for hyperbolic equations subject to general integral conditions is one of the rarely studied problems of mathematical physics This formulation of the problem is considered for the first time.

The description of the method and the algorithm

Conditions for convergence of the algorithm and the main result