On One Solution of a Nonlocal Boundary Value Problem for a Nonlinear Partial Differential Equation of the Third Order

In this paper, a nonlocal boundary value problem for the Benjamin-Bona-Mahony-Burgers equation is studied in a rectangular domain. By introducing new functions, the nonlocal boundary value problem for a nonlinear third-order partial differential equation is reduced to a boundary value problem for a second-order hyperbolic equation with a mixed derivative and functional relations. Before using the approximate method, the nonlinear problem under consideration is examined for the presence of solutions, it is necessary to clarify where these solutions are located, that is, to find the region of isolation of solutions. The isolation area of the solution in our case is a ball in which there is a unique solution to the problem. Next, an algorithm for finding a solution to a nonlocal boundary value problem is proposed. In terms of the initial data, conditions for the convergence of the algorithms are established, which simultaneously ensure the existence and isolation of a solution to a nonlinear nonlocal boundary value problem. Estimates between the exact and approximate solutions of the problem under consideration are obtained. The results obtained are of a theoretical nature and can be used in the construction of computational algorithms for solving nonlocal boundary value problems for the Benjamin-Bona-Mahony-Burgers equation.