Study Of Boundary Value Problems Research Articles

Investigation of the solvability of boundary value problems for the nonlocal Poisson equation with periodic conditions in circular domains

In this paper, boundary value problems with transformed arguments are studied in the unit ball. The transformation of the arguments is specified using the involution type mapping. These mappings participate both in the equation and in the boundary conditions. The equation under consideration is a nonlocal analog of the Poisson equation. Boundary conditions are specified as a relationship between the value of the desired function in the upper hemisphere and the value of the lower hemisphere. These conditions generalize the known periodic conditions for spherical regions. When studying boundary value problems, the properties of involutive mappings are used. The problems under consideration are solved by reducing them to analogues of boundary value problems with periodic conditions for the classical Poisson equation. Using well-known statements for periodic problems for the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Exact conditions for the solvability of the problems under study are found. Spectral questions related to periodic problems are also studied. Eigenfunctions and eigenvalues of these problems are found.

A boundary value problem for the fourth-order degenerate equation of the mixed type

Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary value problems for mixed type’s equations of the fourth-order with power-law degeneration remains relevant. In this work, a boundary value problem in a rectangular domain for a degenerate equation of the fourth-order mixed-type is posed and investigated. Well-posedness of the boundary value problem for a fourth-order partial differential equation is established by proving the existence and uniqueness of the solution. Under sufficient conditions, a solution to the problem under consideration was explicitly found by the variable separation method.

Some new methods for studying boundary value problems for general partial differential equations

Some new methods for studying boundary value problems for general partial differential equations

Some new methods for studying boundary value problems for general partial differential equations

Рассматриваются методы изучения граничных задач для линейных дифференциальных уравнений с частными производными в области независимо от типа уравнения. Предлагается несколько методов изучения граничных задач, которые, как правило, основываются на формуле Грина. Этим методам были посвящены ранее вышедшие публикации автора, а в настоящей статье эти результаты представлены в собранном виде и в краткой форме.

Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Let $\Omega \subset \mathbb {R}^N$ ( $N\geq 3$ ) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$ , $0\leq k \leq N-1$ . We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$ . Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$ , where $\mu \in {\mathbb {R}}$ . We study boundary value problems ( $P_\pm$ ) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$ , where $p>1$ , $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$ . Different critical exponents for the existence of a solution to ( $P_\pm$ ) appear according to concentration of $\nu$ . The solvability for problem ( $P_+$ ) was proved in [3, 29] in subcritical ranges for $p$ , namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$ , we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ( $P_+$ ) in supercritical ranges for $p$ , i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ( $P_-$ ) under a smallness assumption on $\nu$ .

A new algorithm of boundary value problems based on improved wavelet basis and the reproducing kernel theory

Relying upon Legendre polynomials, an improved multiwavelet orthonormal basis is constructed in the reproducing kernel space. We study boundary value problems (BVPs) based on the improved wavelet orthonormal basis and construct the ‐best approximate solution. Convergence and the stability are discussed; this method has higher order convergence. The main advantage of the wavelet based our method is its efficiency and simple applicability for a variety of boundary conditions. Some numerical tests even including singular BVPs reveal the stability and efficiency of the new algorithm.

Boundary value problems on non-Lipschitz uniform domains: stability, compactness and the existence of optimal shapes

We study boundary value problems for bounded uniform domains in R n , n ⩾ 2, with non-Lipschitz, and possibly fractal, boundaries. We prove Poincaré inequalities with uniform constants and trace terms for ( ε , ∞ )-domains contained in a fixed bounded Lipschitz domain. We introduce generalized Dirichlet, Neumann, and Robin problems for Poisson-type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, the norm convergence of the associated resolvents, and the convergence of the corresponding eigenvalues and eigenfunctions. We provide compactness results for parametrized classes of admissible domains, energy functionals, and weak solutions. Using these results, we can then prove the existence of optimal shapes in these classes in the sense that they minimize the initially given energy functionals. For the Robin boundary problems, this result is new.

To the Properties of One Fox Function

В работе рассматривается частный случай специальной функции Фокса с четырьмя параметрами, которая возникает в теории краевых задач для параболических уравнений c оператором Бесселя и дробной производной по времени. Целью исследования является получение некоторых рекуррентных соотношений, формул дифференцирования и интегрального преобразования рассматриваемой функции. При получении результатов работы в основном используется представление рассматриваемой функции через интеграл Меллина–Барнса. Также используются её асимптотические разложения при большом и малом значениях аргумента. С помощью указанного интегрального представления и некоторых известных формул для гамма-функции Эйлера, получены рекуррентные соотношения, связывающие функции с разными параметрами, а также функцию с её производной первого порядка. Получена формула дифференцирования n-го порядка. Исследуется несобственный интеграл первого рода, который содержит рассматриваемую функцию с двумя зависимыми параметрами из четырёх. Показывается, что этот несобственный интеграл может быть записан в терминах известной специальной функции Макдональда. При частных значениях параметров рассматриваемой в работе функции получаются некоторые известные элементарные и специальные функции. Результаты работы носят теоретический характер и будут полезны при исследовании краевых задач для вырождающихся параболических уравнений с производными дробного порядка по времени. The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin–Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the nth order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.

Double Sawi Transform: Theory and Applications to Boundary Values Problems

Symmetry can play an important role in the study of boundary value problems, which are a type of problem in mathematics that involves finding the solutions to differential equations subject to given boundary conditions. Integral transforms play a crucial role in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and integral equations. This article focuses on extending a single-valued Sawi transform to a double-valued ST, which we call the double Sawi (DS) transform. We derive some fundamental features and theorems for the proposed transform. Finally, we study the applications of the proposed transform by solving some boundary value problems such as the Fourier heat equation and the D’Alembert wave equation.

Solvability of a mixed problem with the integral gluing condition for a loaded equation with the Riemann–Liouville fractional operator

In this paper, we study boundary value problems with an integral gluing condition for a loaded equation of parabolic–hyperbolic type. The existence and uniqueness of the problem under study are proved based on the unique solvability obtained from integral and loaded integral equations.

On the unique solvability of a nonlocal boundary value problem with the poincaré condition

As is known, it is customary in the literature to divide degenerate equations of mixed type into equations of the first and second kind. In the case of an equation of the second kind, in contrast to the first, the degeneracy line is simultaneously the envelope of a family of characteristics, i.e. is itself a characteristic, which causes additional difficulties in the study of boundary value problems for equations of the second kind. In this paper, in order to establish the unique solvability of one nonlocal problem with the Poincaré condition for an elliptic-hyperbolic equation of the second kind developed a new principle extremum, which helps to prove the uniqueness of resolutions as signed problem. The existence of a solution is realized by reducing the problem posed to a singular integral equation of normal type, which known by the Carleman-Vekua regularization method developed by S.G. Mikhlin and M.M. Smirnov equivalently reduces to the Fredholm integral equation of the second kind, and the solvability of the latter follows from the uniqueness of the solution delivered problem.

Dirichlet’s problem for a third-order parabolic hyperbolic type equation of the second kind

Equations of mixed type, the degeneracy line of which is the envelope of a family of characteristics, therefore, is itself also a characteristic, in the literature it is customary to call equations of mixed type of the second kind, which causes additional difficulties in the study of boundary value problems for equations of the second kind. In the present work, a boundary value problem for a homogeneous equation of parabolic-hyperbolic type of the third order of the second kind is investigated. Necessary and sufficient conditions for the existence and uniqueness of a generalized solution of the problem are found. In some special cases, the representation of the solution of the problem under study is written out explicitly.

Existence and uniqueness results for fractional boundary value problems with multiple orders of fractional derivatives and integrals

In this article, we study boundary value problems for fractional differential equations with multiple orders of fractional derivatives and integrals, in both fractional differential equation and boundary conditions. Existence and uniqueness results are obtained using Sadovskii’s fixed point theorem. Finally, numerical examples are provided to confirm the applicability of obtained theoretical results.

Two theorems on estimates for solutions of one class of nonlinear equations in a finite-dimensional space

The need to study boundary value problems for elliptic parabolic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory and quantum physics. In this paper, we obtain two theorems on a priori estimates for solutions of nonlinear equations in a finite-dimensional Hilbert space. The work consists of four items. In the first subsection, the notation used and the statement of the main results are given. In the second subsection, the main lemmas are given. The third section is devoted to the proof of Theorem 1. In the fourth section, Theorem 2 is proved. The conditions of the theorems are such that they can be used in studying a certain class of initial-boundary value problems to obtain strong a priori estimates in the presence of weak a priori estimates. This is the meaning of these theorems.

Boundary value problems for general first-order elliptic differential operators

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.Link to video abstract: https://vimeo.com/523211595

Multi-Point Boundary Value Problems for (k, ϕ)-Hilfer Fractional Differential Equations and Inclusions

In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving (k,ϕ)-Hilfer fractional derivative of order in (1,2]. In the single-valued case the existence and uniqueness results are established by using classical fixed-point theorems, such as Banach, Krasnoselskiĭ and Leray-Schauder. In the multivalued case we consider both cases, when the right-hand side has convex or non-convex values. In the first case, we apply the Leray–Schauder nonlinear alternative for multivalued maps, and in the second, the Covit–Nadler fixed-point theorem for multivalued contractions. All results are well illustrated by numerical examples.

On the solutions of fractional integro-differential equations involving Ulam-Hyers-Rassias stability results via $\psi$-fractional derivative with boundary value conditions

In this paper, we study boundary value problems for the impulsive integro-differential equations via $\psi$-fractional derivative. The contraction mapping concept and Schaefer's fixed point theorem are used to produce the main results. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam-Hyers-Rassias stability of the solution to the proposed system.

Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Equation with Iterated Fractional Integrals Conditions

In the present research, we initiate the study of boundary value problems for sequential Riemann–Liouville and Hadamard–Caputo fractional derivatives, supplemented with iterated fractional integral boundary conditions. Firstly, we convert the given nonlinear problem into a fixed point problem by considering a linear variant of the given problem. Once the fixed point operator is available, we use a variety of fixed point theorems to establish results regarding existence and uniqueness. Some properties of iteration that will be used in our study are also discussed. Examples illustrating our main results are also constructed. At the end, a brief conclusion is given. Our results are new in the given configuration and enrich the literature on boundary value problems for fractional differential equations.

Difference formula defined by a new differential symmetric operator for a class of meromorphically multivalent functions

Symmetric operators have benefited in different fields not only in mathematics but also in other sciences. They appeared in the studies of boundary value problems and spectral theory. In this note, we present a new symmetric differential operator associated with a special class of meromorphically multivalent functions in the punctured unit disk. This study explores some of its geometric properties. We consider a new class of analytic functions employing the suggested symmetric differential operator.

Boundary value problems for Hilfer type sequential fractional differential equations and inclusions involving Riemann\u2013Stieltjes integral multi-strip boundary conditions

In this paper, we study boundary value problems for sequential fractional differential equations and inclusions involving Hilfer fractional derivatives, supplemented with Riemann–Stieltjes integral multi-strip boundary conditions. Existence and uniqueness results are obtained in the single-valued case by using the classical Banach and Krasnosel’skiĭ fixed point theorems and the Leray–Schauder nonlinear alternative. In the multi-valued case an existence result is proved by using nonlinear alternative for contractive maps. Examples illustrating our results are also presented.