On the non-local boundary value problem of a third order hyperbolic equation

Как известно, в работе А.В. Бицадзе показано, что задача Дирихле для уравнения смешанного типа некорректна. Естественно возникает вопрос: нельзя ли заменить условия задачи Дирихле другими условиями, охватывающими всю границу, которые обеспечивают корректность задачи? Впервые такие краевые задачи (нелокальные краевые задачи) для уравнения смешанного типа были предложены и изучены в работах Ф.И. Франкля при решении газодинамической задачи об обтекании профилей потоком дозвуковой скорости со сверхзвуковой зоной, оканчивающейся прямым скачком уплотнения. Близкие по постановке задачи для уравнения смешанного типа второго рода второго порядка, имеются в работах А.Н. Терехова, С.Н. Глазатова, М.Г. Каратопраклиевой и С.З. Джамалова. В этих работах для уравнения смешанного типа второго рода второго порядка изучены нелокальные краевые задачи в ограниченных областях. Такие задачи для уравнения смешанного типа первого рода в трехмерном случае (в частости, для уравнения Трикоми) в неограниченных областях изучены в работах С.З. Джамалова и Х. Туракулова. Для уравнений смешанного типа второго рода в неограниченных областях нелокальные краевые задачи в многомерном случае практически не исследованы. С этой целью в данной работе в неограниченном параллелепипеде формулируется и изучается нелокальная краевая задача периодического типа для трехмерного уравнения смешанного типа второго рода второго порядка. Для доказательства единственности обобщённого решения используется метод интегралов энергии. Для доказательства существования обобщённого решения сначала используется преобразование Фурье и в результате получается новая задача на плоскости, а для разрешимости этой задачи используется методы «ε-регуляризации»и априорных оценок. Используя эти методы, и равенство Парсеваля, докажем единственность, существование и гладкость обобщённого решения одной нелокальной краевой задачи периодического типа для трехмерного уравнения смешанного типа второго рода второго порядка. As is known, A.V. Bitsadze in his studies pointed out that the Dirichlet problem for a mixed-type equation, in particular for a degenerate hyperbolic-parabolic equation, is ill-posed. The question naturally arises: is it possible to replace the conditions of the Dirichlet problem with other conditions covering the entire boundary, which will ensure the well-posedness of the problem? For the first time, such boundary value problems (nonlocal boundary value problems) for a mixed-type equation were proposed and studied in the works of F.I. Frankl when solving the gas-dynamic problem of subsonic flow around airfoils with a supersonic zone ending in a direct shock wave. Problems close in formulation to a mixed-type equation of the second order were considered in the studies by A.N. Terekhov, S.N. Glazatov, M.G. Karatopraklieva and S.Z. Dzhamalov. In these papers, nonlocal boundary value problems in bounded domains are studied for a mixed-type equation of the second kind of the second order. Such problems for a mixed-type equation of the first kind in the three-dimensional case (in particular, for the Tricomi equation) in unbounded domains are studied in the works of S.Z. Dzhamalov and H. Turakulov. For mixed-type equations of the second kind in unbounded domains, nonlocal boundary value problems in the multidimensional case are practically not studied. In this article, nonlocal boundary value problem of periodic type for a mixed-type equation of the second kind of the second order, is formulated and studied in an unbounded parallelepiped. To prove the uniqueness of the generalized solution, the method of energy integrals is used. To prove the existence of a generalized solution, the Fourier transforms is used and as a result, a new problem is obtained on the plane. And for the solvability of this problem, the methods of “ε-regularization”and a priori estimates are used. The uniqueness, existence, and smoothness of a generalized solution of a nonlocal boundary value problem of periodic type for a three-dimensional mixed-type equation of the second kind of the second order are proved using above-mentioned methods and Parseval equality.

In the present article we are interested in the analysis of nonlocal initial boundary value problems for some medium oscillation equations. More precisely, we investigate different types of nonlocal problems for one-dimensional oscillation equations and prove existence and uniqueness theorems. In some cases algorithms for direct construction of the solution are given. We also consider nonlocal problem for multidimensional hyperbolic equation and prove the uniqueness theorem for the formulated initial boundary value problem applying the theory of characteristics under rather general assumptions.

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.

Nonlocal conjugate type boundary value problems of higher order

Pseudo-differential operators and -toroidal symbols generated by a non-local boundary value problem are investigated. A formula for compositions with pseudo-differential operators generated by a non-local boundary value problem is derived. In the Hilbert space concepts of the Fourier transform and convolution generated by a non-local boundary value problem are introduced.

In the article we study the questions of well-posedness of general nonlocal boundary value problems for pseudodifferential equations in the Besov-type limit spaces.

This paper is devoted to a Bitsadze-Samarskii type overdetermined multipoint nonlocal boundary value problem (NBVP). This inverse problem is reduced to an auxiliary multipoint NBVP. Stability estimates for the solution of the auxiliary NBVP are established. The finite difference method is applied to get the first and second order of accuracy of approximate solutions of the abstract overdetermined problem. Stability estimates for the solution of difference problems are proved. Then the established abstract results are applied to get stability estimates for the solution of the Bitsadze-Samarskii type overdetermined elliptic multidimensional differential and difference problems with multipoint nonlocal boundary conditions (NBVC). Finally, numerical results with explanation on the realization for two dimensional and three dimensional elliptic overdetermined multipoint NBVPs in test examples are presented.

We consider nonlocal boundary value problems for a Sobolev-type equation with variable coefficients with fractional Gerasimov–Caputo derivative. The main result of the article consists in proving a priori estimates for solutions to nonlocal boundary value problems both in differential and difference form obtained under the assumption of the existence of a solution u(x, t) in a class of sufficiently smooth functions. These inequalities imply the uniqueness and stability of a solution with respect to the initial data and right-hand side and also the convergence of the solution to the difference problem to the solution to the differential problem.

We consider a nonlocal boundary value problem for a third order differential equation. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. The results are illustrated with some examples.

ABSTRACTIn this paper, we study a non-local initial boundary value problem for a one-dimensional pseudohyperbolic equation. We first establish the existence and uniqueness of strong solution, then a numerical solutions for the system will be derived by using the finite-difference method.

Nonlocal and multiple-point boundary value problem for fractional differential equations

A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze‐Samarskii‐Dirichlet condition. The first and second‐orders of accuracy stable difference schemes for the approximate solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity, and almost coercivity inequalities for solution of these schemes are established. The theoretical statements for the solutions of these nonlocal elliptic problems are supported by results of numerical examples.

We consider a nonlocal boundary-value problem with impulsive actions for a system of loaded hyperbolic equations and establish the relationship between the unique solvability of this problem and the unique solvability of a family of two-point boundary-value problems with impulsive actions for the system of loaded ordinary differential equations by the method of introduction of additional functions. By the method of parametrization, we establish sufficient conditions for the existence of a unique solution to a family of two-point boundary-value problems with impulsive effects for a system of loaded ordinary differential equations. The algorithms of finding the solutions of these boundary-value problems are constructed. The conditions for the unique solvability of the nonlocal boundary-value problem for a system of loaded hyperbolic equations with impulsive actions are obtained. We also propose the numerical realization of the algorithms of the method of parametrization for the solution of the family of two-point boundary-value problems with impulsive actions for the system of loaded ordinary differential equations. The results are illustrated by specific examples.

This study discussed the existence of solutions for a nonlinear fractional integro-differential equation of order with four-point nonlocal integral boundary conditions. The given problem is transformed to an equivalent fixed point problem in terms of an operator equation. Then, by means of Banach contraction principle and a fixed point theorem due to Krasnoselskii, the existence results are obtained. The last existence result is based on nonlinear alternative of Leray-Schauder type. An illustrative example is also presented. Key words: Integro-differential equations, fractional order, four-point integral boundary conditions, fixed point theorems, Leray-Schauder degree.

PurposeThe purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.Design/methodology/approachThe authors first transform the given nonlocal boundary value problems of first‐ and second‐order differential equations into local boundary value problems of second‐ and third‐order differential equations, respectively. Then a modified Adomian decomposition method is applied, which permits convenient resolution of these equations.FindingsThe new technique, as presented in this paper in extending the applicability of the Adomian decomposition method, has been shown to be very efficient for solving nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.Originality/valueThe paper presents a new solution algorithm for the nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.