The thesis is a continuation of research in the field of constructive methods for approximate solution of boundary value problems and is dedicated to the development of new modifications of the two-sided accelerated convergence method for the investigation and approximate solution of boundary value problems for nonlinear partial differential equations in domains with complex boundary structure, pre-history, and problems with non-local boundary conditions.
The research task is to build constructive two-sided methods for the approximate solution of boundary value problems for quasilinear wave equations with a discontinuous right side in a domain with a complex boundary structure, boundary value problems for nonlinear wave equations in a domain with a complex boundary structure and a pre-history, as well as problems with a non-local boundary condition of Nahushew A.M. for a system of quasilinear partial differential equations; to find sufficient conditions for the existence of regular and irregular solutions of the investigated problem, their uniqueness, and stability.
The introduction substantiates the relevance of the conducted research, formulates the aim and objectives, specifies the object, subject, and research methods, indicates the scientific novelty and practical significance of the obtained results.
In the first chapter, a review of literature sources and the main tasks related to the topic of the dissertation research has been conducted.
The second chapter is dedicated to boundary value problems for wave equations with discontinuous right sides in domains with complex boundary structure. The investigated boundary problem is reduced to a system of equivalent integral equations. One fast-converging modification of the two-sided method for its approximate solution has been constructed, establishing sufficient conditions for the existence of a regular or irregular solution to the boundary value problem, as well as its uniqueness and sign stability. It is demonstrated that the set of acceleration functions is non-empty. Practical methods for constructing comparison functions for boundary value problems have been provided. An a posteriori estimate of the approximate solution to the investigated problem has been obtained. The uniform convergence of the constructed sequences of functions to a unique continuous solution of the system of integral equations has been proven. Additionally, another approach to approximate solution of the boundary value problem is presented in this chapter, illustrated with an example.
The third chapter investigates boundary value problems for nonlinear wave equations in domains with complex boundary structure and pre-history. The original boundary value problem is reduced to an equivalent system of nonlinear integral equations, for which a constructive fast-converging two-sided method for investigation and approximate solution has been constructed. Sufficient conditions for the existence and uniqueness of the solution to the system of integral equations have been established. The uniform convergence of the constructed sequences of functions to a unique solution of the system of integral equations has been proven. It`s been established that the set of acceleration functions for convergence is non-empty. Sufficient conditions for the regularity or irregularity of the solution of the investigated problem and its sign stability have been proven. A practical method for finding comparison functions for the boundary value problem has been demonstrated.
In the fourth chapter, problems with a non-local boundary condition of A.M. Nakhushev have been investigated in the case of a system of quasi-linear partial differential equations. One modification of the two-sided accelerated convergence method for their approximate solution has been proposed. A theorem on the uniform convergence of the constructed sequences of functions to a unique solution of the boundary value problem has been proven. It`s been shown that the convergence of the constructed method is not slower than the convergence of previously known methods. A theorem on differential inequalities has been proven, and an a posteriori error estimate for the approximate solution has been obtained. Additionally, in this chapter, an illustrative example has been provided, in which sequences of functions for two, along with analysis of the obtained posterior estimates of approximations have been conducted.
The results obtained in the scientific paper have both theoretical and practical significance. They generalize and complement previous research in qualitative theory of boundary value problems and constructive methods theory. The developed modifications of two-sided methods can be applied to solve practical problems, the mathematical models of which are boundary value problems for nonlinear partial differential equations in domains with complex boundary structure, pre-history, and non-local boundary conditions.
