Nonlocal Boundary Value Problem Research Articles

On a fully coupled nonlocal multipoint boundary value problem for a dual hybrid system of nonlinear q-fractional differential equations

On a fully coupled nonlocal multipoint boundary value problem for a dual hybrid system of nonlinear <i>q</i>-fractional differential equations

Nonlocal boundary value problems for functional hybrid differential equations involving generalized w-Caputo fractional operator

In this manuscript , we establish the existence and uniqueness of solutions for boundary value problems of nonlinear hybrid fractional differential equations involving generalized $\omega-$Caputo fractional derivatives. The proofs are based on Krasnoselskii fixed point theorem and some basic concept of $\omega-$Caputo fractional analysis. As application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.

On strong solvability of one nonlocal boundary value problem for Laplace equation in rectangle

On strong solvability of one nonlocal boundary value problem for Laplace equation in rectangle

Numerical approaches for solution of hyperbolic difference equations on circle

Abstract The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 \mathbb{T}^{1} . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.

Stability on a boundary problem with RL-Fractional derivative in the sense of Atangana-Baleanu of variable-order

&lt;abstract&gt;&lt;p&gt;In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable.&lt;/p&gt;&lt;/abstract&gt;

Positive solutions to a third order nonlocal boundary value problem with a parameter

Positive solutions to a third order nonlocal boundary value problem with a parameter

Existence of solutions for fractional boundary value problems with Riesz space derivative and nonlocal conditions

"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. "

On ellipticity of operators with shear mappings

The nonlocal boundary value problems are considered, in which the main operator and the operators in the boundary conditions include the differential operators and twisting operators. The de nition of the trajectory symbols for this class of problems is given. We show that the elliptic problems de ne the Fredholm operators in the corresponding Sobolev spaces. The ellipticity condition of such nonlocal boundary value problem is given.

Semiclassical analysis of a nonlocal boundary value problem related to magnitude

AbstractWe study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy. Using recent techniques developed for pseudodifferential boundary problems we discuss the structure of the solution operator and resulting properties of the magnitude. In a semiclassical limit we obtain an asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and the boundary, similar to the invariants arising in short-time expansions for heat kernels.

A matrix technique‐based numerical treatment of a nonlocal singular boundary value problems

The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundary value problems. The nonlocal boundary conditions are more informative than local boundary conditions (initial conditions and two‐point boundary conditions) to evaluate some mathematical models. This article presents a collocation approach‐based matrix technique to approximate the solution of the fusion of a class of singular differential equations subject to nonlocal three‐point boundary conditions. The proposed strategy utilizes the truncation of the series expansion of a function belonging to in terms of Bernoulli polynomials. It transforms the singular boundary value problems into a set of nonlinear algebraic equations, which can be dealt with by any mathematical software. The Lipschitz condition on an equivalent completely continuous nonlinear operator has been used to prove the convergence analysis of the scheme. Some extremely nonlinear test examples are solved and provided in contrast with the exact solution. These numerical results are also examined against some existing numerical techniques to verify the applicability and significance of the proposed methodology. There are a few numerical examples that are application based but do not have exact solutions. In such cases, residual error norm is employed to measure the accuracy of the numerical strategies. The computational data demonstrate the superiority and validity of the proposed technique over existing numerical approaches.

A generalized integral problem for a system of hyperbolic equations and its applications

A nonlocal boundary value problem for a system of hyperbolic equations of second order with generalized integral condition is considered. By method of introduction of functional parameters the investigated problem is transformed to the inverse problem for the system of hyperbolic equations with unknown parameters and additional functional relations. Algorithms of finding solution to the inverse problem for the system of hyperbolic equations are constructed, and their convergence is proved. The conditions for existence of unique solution to the inverse problem for the system of hyperbolic equations are obtained in the terms of initial data. The coefficient conditions for unique solvability of nonlocal boundary value problem for the system of hyperbolic equations with generalized integral condition are established. The results are illustrated by numerical examples.

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

This research work deals with two spectral matrix collocation algorithms based on (novel) clique functions to solve two classes of nonlinear nonlocal elliptic two-points boundary value problems (BVPs) arising in diverse physical models. The nonlinearity together with nonlocality makes the models so difficult to solve numerically. In both matrix methods by expanding the unknown solutions in terms of clique polynomials the nonlocality in the models is handled. In the first and direct technique, the clique coefficients are determined in an accurate way through solving an algebraic system of nonlinear equations. To get rid of the nonlinearity of the models and in order to gain efficacy, the quasilinearization method (QLM) is utilized in the second approach. In the latter algorithm named QLM-clique procedure, the first directly clique collocation method is applied to a family of linearized equations in an iterative manner. In particular, we show that the series expansion of clique polynomials is convergent exponentially in a weighted L2 norm. Numerous numerical simulations validate the performance of two matrix collocation schemes and demonstrate the accuracy as well as the gain in computational efficiency in terms of elapsed CPU time. The proposed matrix algorithms for computation are easy to implement and straightforward, and provide more accuracy compared to other available computational values in the literature.

BOUNDARY VALUE PROBLEMS WITH AN INTEGRO-DIFFERENTIAL NON-LOCAL CONDITION FOR COMPOSITE TYPE DIFFERENTIAL EQUATIONS OF THE FOURTH ORDER

The paper studies new nonlocal boundary value problems with an integro-differential boundary condition for unsteady differential equations of the Sobolev type of the fourth order. The peculiarity of the studied problems is that they contain derivatives both in spatial variables and derivatives in time variables in the boundary condition. For the problems under study, the existence and uniqueness theorems of regular solutions are proved – solutions having all derivatives generalized by S.L. Sobolev included in the corresponding equations.

Численные методы решения нелокальных краевых задач для обобщенных нагруженных уравнений Аллера

Работа посвящена нелокальным краевым задачам для одномерных по пространству нагруженных уравнений Аллера с переменными коэффициентами и двумя операторами дробного дифференцирования Капуто с порядками $\alpha$ и $\beta$. Подобные задачи возникают в практике регулирования солевого режима почв, когда расслоение верхнего слоя достигается сливом слоя воды с~поверхности затопленного на некоторое время участка. Для численного решения поставленных задач на равномерной сетке построены разностные схемы. Методом энергетических неравенств при различных соотношениях между порядками дробной производной Капуто $\alpha$ и $\beta$ получены априорные оценки в дифференциальной и разностной трактовках для решений нелокальных краевых задач. Из~полученных априорных оценок следуют единственность и устойчивость решения по правой части и начальным данным, а также сходимость решения разностной задачи к решению соответствующей исходной дифференциальной задачи (в предположении существования решения дифференциальной задачи в классе достаточно гладких функций) со скоростью $O(h^2+\tau^2)$ при $\alpha=\beta$ и $O(h^2+\tau^{2-\max\{\alpha,\beta\}})$ при $\alpha\neq\beta$. В работе также приводится алгоритм численного решения нелокальной краевой задачи для нагруженного уравнения Аллера с переменными коэффициентами и оператором Бесселя.

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.

Non-local fractional boundary value problems with applications to predator-prey models

We study a nonlinear fractional boundary value problem (BVP) subject to non-local multipoint boundary conditions. By introducing an appropriate parametrization technique we reduce the original problem to an equivalent one with already two-point restrictions. Using a notion of Chebyshev nodes and Lagrange polynomials we construct a successive iteration scheme, that converges to the exact solution of the non-local problem for particular values of the unknown parameters, which are calculated numerically.&#x0D; For mote information see https://ejde.math.txstate.edu/Volumes/2023/58/abstr.html&#x0D;

Linear inverse problems for the heat equation and non-local boundary value problems with generalized Samarskii–Ionkin condition

Linear inverse problems for the heat equation and non-local boundary value problems with generalized Samarskii–Ionkin condition

Nonlocal Boundary Value Problem for a Mixed Type Equation with Fractional Partial Derivative

Nonlocal Boundary Value Problem for a Mixed Type Equation with Fractional Partial Derivative

On a Solution of a Nonlinear Nonlocal Boundary Value Problem for one Class of Hyperbolic Equation

On a Solution of a Nonlinear Nonlocal Boundary Value Problem for one Class of Hyperbolic Equation

On the Weak Solvability of a Nonlocal Boundary Value Problem for the Laplace Equation in an Unbounded Domain

On the Weak Solvability of a Nonlocal Boundary Value Problem for the Laplace Equation in an Unbounded Domain