Nonlocal Boundary Conditions Research Articles

Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions

<abstract><p>This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results.</p></abstract>

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

О РЕШЕНИИ НЕЛОКАЛЬНОЙ ОБРАТНОЙ ЗАДАЧИ ДЛЯ ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ

The article investigates a problem with inverse time for a semilinear parabolic equation equipped with nonlocal boundary conditions. The problem under study arises, for example, in the mathematical modeling of the process of external gettering of silicon wafers in the creation of semiconductor devices. At the same time, when developing a mathematical model in the case of a high-intensive diffusion process, it is necessary to take into account the effects associated with the nonlinearity of the process. The paper suggests an approach for constructing a numerical solution to the problem with the inverse time, stable with respect to small perturbations of the initial data. The numerical solution is constructed using the regularization method based on adding a term with a small parameter to the overdetermination (final) condition. To obtain an approximate solution, the problem statement must recon in supplementary (a priori) information characterizing the exact solution. We obtain an error estimate for the approximate solution under the given a priori information.

Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions

<abstract><p>Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.</p></abstract>

Extended existence results for FDEs with nonlocal conditions

<abstract><p>This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $ 0\leq c < d $ and fixed points at the end of the interval $ [0, d] $. For $ i = 0, 1 $, the boundary conditions are as follows: $ a_{i}, b_{i} > 0 $, $ a_{0} p(c) = -b_{0} p(d), \ a_{1} p^{'}(c) = -b_{1} p^{'}(d) $. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.</p></abstract>

Correction to: On Solvability of a Linear Parabolic Problem with Nonlocal Boundary Conditions

Correction to: On Solvability of a Linear Parabolic Problem with Nonlocal Boundary Conditions

Existence and Uniqueness of a Solution for a Four‐Stage Age‐Structured Population Dynamics Model With Spatial Diffusion for Desert Locusts

We consider a linear system based on a population dynamics model dependent on age, space and nonlocal boundary conditions. Note that our population dynamics model is a four‐step model with a second derivative with respect to the age variable and a second derivative with respect to the space variable. Population growth at each stage depends on time and space. We prove the uniqueness and existence of a positive solution by combining the Galerkin’s variational method and the Banach’s fixed point theorem.

On the solutions of the second-order $ (p, q) $-difference equation with an application to the fixed-point theory

<abstract><p>In this paper, we examined the existence and uniqueness of solutions to the second-order $ (p, q) $-difference equation with non-local boundary conditions by using the Banach fixed-point theorem. Moreover, we introduced a special case of this equation called the Euler-Cauchy-like $ (p, q) $-difference equation and provide its solution. We also studied the oscillation of solutions for this equation in $ (p, q) $-calculus and proved the $ (p, q) $-Sturm-type separation theorem and $ (p, q) $-Kneser theorem about the oscillation of solutions.</p></abstract>

On Solvability of a Linear Parabolic Problem with Nonlocal Boundary Conditions

On Solvability of a Linear Parabolic Problem with Nonlocal Boundary Conditions

Exact controllability for nonlocal wave equations with nonlocal boundary conditions

Exact controllability for nonlocal wave equations with nonlocal boundary conditions

ДЕЯКІ КРАЙОВІ ЗАДАЧІ, ВІДПОВІДНІ МОДЕЛІ ДРОБОВО-ДИФЕРЕНЦІЙНОЇ ФІЛЬТРАЦІЙНОЇ ДИНАМІКИ В ТРІЩИНУВАТО-ПОРИСТОМУ СЕРЕДОВИЩІ ЗА УМОВ ЧАСОВОЇ НЕЛОКАЛЬНОСТІ

Closed-form solutions of some boundary-value problems of fractional-differential geofiltration dynamics in a fractured-porous medium are obtained for a model with weakly permeable porous blocks. In particular, the direct and inverse boundary-value problems of filtration for the finite thickness layer are solved, the conditions for the existence of their regular solutions are given, and the solution of the problem of filtration dynamics with nonlocal boundary conditions is found. For a particular case of the filtration model, the problem of modeling the anomalous dynamics of filtration pressure fields on a star-shaped graph is considered. Keywords: mathematical modeling, fractional-differential dynamics of filtration processes, fractured-porous medium, non-classical models, boundary-value problems, inverse problems, problems with non-local conditions, closed-form solutions.

Existence and uniqueness results for a class of fractional differential equations with nonlocal boundary conditions

Existence and uniqueness results for a class of fractional differential equations with nonlocal boundary conditions

Unknown coefficient problem for mixed equation of parabolic-hyperbolic type with non-local boundary conditions on characteristics

For an equation of a mixed parabolic-hyperbolic type with a characteristic line of type change, we study the inverse problem associated with the search for an unknown coefficient at the lowest term of the parabolic equation. In the direct problem, we consider an analog of the Tricomi problem for this equation with a nonlocal condition on the characteristics in the hyperbolic part and the Dirichlet condition in the parabolic part of the domain. In order to determine the unknown coefficient by the solution on the parabolic part of the domain, the integral overdetermination condition is proposed. Global results on the unique solvability of the inverse problem in the sense of the classical solution are proved.

On a Linear Fractional Differential Equation Involving Liouville Derivative

The paper investigates linear fractional differential equations involving the Liouville derivative. Solution to these equations under a non-local boundary condition are derived in explicit form, their uniqueness is established using integral transforms technique.

Inverse source problem for multi-term time-fractional diffusion equation with nonlocal boundary conditions

<abstract><p>In this paper, we consider an inverse source problem with nonlocal boundary conditions for the heat equation involving multi-term time-fractional derivatives. We determine a source term independent of the space variable, and the temperature distribution from the energy measurement. We reduce the solution of the inverse problem to finding solutions to two problems. The well-posedness of each problem is shown using the generalized Fourier method.</p></abstract>

Utilizing Schaefer's fixed point theorem in nonlinear Caputo sequential fractional differential equation systems

<abstract><p>In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.</p></abstract>

A MULTIPOINT BOUNDARY VALUE PROBLEM IN TIME FOR A $2B$-PARABOLIC EQUATION WITH DEGENERACY

One of the most important issues in the general theory of differential equations with partial derivatives is establishing the solvability of boundary value problems. Among the boundary value problems for equations with partial derivatives, problems with nonlocal boundary conditions occupy an important place. Such interest in such problems is caused both by their rich practical application (the process of diffusion, moisture distortion in soils, plasma physics, etc.), and by the needs of the general theory of boundary value problems. A general multipoint boundary value problem for nonuniformly $2b$-parabolic equations with degeneracy is studied. The coefficients of parabolic equations and boundary conditions allow power degeneracy of arbitrary order in terms of time variable and spatial variables at some set of points. To solve the given multipoint boundary value problem, solutions of problems with smooth coefficients in Hölder spaces with the appropriate norm are studied. With the help of interpolation inequalities and a priori estimates, estimates of the solution of auxiliary problems and their derivatives in special Gelder spaces are established. Using the theorems of Ross and Archel, a convergent sequence is distinguished from the compact sequence of solutions of the auxiliary problems, the limiting value of which is the solution of the multipoint boundary value problem in time for the $2b$-parabolic equation with degeneracy. Estimates of the solution of the given problem are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of features of the coefficients of the equations and the boundary conditions. With certain restrictions on the right-hand side of the equation and boundary conditions, an integral image of the given problem is obtained.

Positive solutions of a nonlinear three-point p-Laplacian fractional boundary value problem with infinitely many singularities

In this paper, an attempt has been made to establish the existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential equation subject to the non-local boundary conditions. We have shown that there exist countably many positive solutions with the help of fixed point index theory in a cone on a Banach space.

Mixed sequential type pantograph fractional integro-differential equations with non-local boundary conditions

Mixed sequential type pantograph fractional integro-differential equations with non-local boundary conditions

The positivity of the differential operator generated by hyperbolic system of equations

In the present paper, the initial value problem for the hyperbolic system of equations with nonlocal boundary conditions is investigated. The positivity of the space operator generated by the given problem in interpolation spaces is established. The structure of interpolation spaces of this differential operator is studied. The positivity of this space operator in Slobodeckij-Sobolev spaces is established.