FDM for Elliptic Equations with Bitsadze‐Samarskii‐Dirichlet Conditions

On existence and uniqueness of solutions of boundary value problems of fourth order elliptic partial differential equations with variable coefficients

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.

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.

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.

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.

The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valuedLp−spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

In this study, we investigate the application of variational methods to establish the existence of weak solutions for boundary value problems (BVPs) associated with nonlinear elliptic partial differential equations (PDEs). By employing the direct method in the calculus of variations, we reformulate the given PDEs as minimisation problems over suitable Sobolev spaces, where weak solutions correspond to critical points of appropriately defined energy functionals. The analytical framework incorporates foundational tools such as Sobolev embeddings, compactness theorems, and properties of coercivity and weak lower semi-continuity to ensure the well-posedness of the variational formulation. We present and prove an existence theorem under well-defined conditions on the nonlinear coefficients of the equations, specifically those involving the Carathéodory property, polynomial growth bounds, monotonicity, and symmetry. These conditions are shown to satisfy the hypotheses of the Fucik-Kufner theorem, which guarantees the existence - and, under convexity assumptions, uniqueness - of weak solutions. To demonstrate the utility of the theoretical results, we apply the theorem to selectednonlinear elliptic problems and verify the fulfilment of all required criteria for the existence of solutions. Through this approach, we provide a rigorous and general framework for analysing nonlinear elliptic BVPs without resorting to classical linearization techniques. The findings confirm that variational methods offer powerful, flexible tools for addressing existence questions in complex PDE systems, with relevance to mathematical modelling in physics, engineering, and applied sciences.

A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

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.

"By using the fixed-point theorems, we give sufficient conditions for the existence and uniqueness of solutions for the nonlocal fractional boundary value problem of nonlinear Riesz-Caputo differential equation. The boundedness assumption on the nonlinear term is replaced by growth conditions or by a continuous function. Finally, some examples are presented to illustrate the applications of the obtained results. Keywords: Fractional boundary value problem, Riesz-Caputo fractional derivative, existence and uniqueness, fixed point, nonlocal conditions. "

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