Nonlocal Boundary Value Problem Research Articles

Existence of weak solutions for nonlocal Dirichlet problems with variable-order fractional kernels

This paper is concerned with the existence and uniqueness of weak solutions for a class of linear and nonlinear nonlocal homogeneous Dirichlet boundary value problems with a truncated variable-order fractional kernel γ ( x , y ) . By the structural features of the kernel γ ( x , y ) , a new variable-order fractional Banach space [ W 0 α ( x , y ) , p ( Ω ′ ) ] d is introduced as the suitable solution space, and its some qualitative properties are established. Then under such a functional framework, based on the variational methods, we prove the existence results for the linear problem with a nonsymmetric kernel case and for the nonlinear problem with a symmetric kernel case, respectively.

On some spectral properties of nonlocal boundary-value problems for nonlinear differential inclusion

UDC 517.9 We study the solutions to the Sturm–Liouville boundary-value problem for a nonlinear differential inclusion with nonlocal conditions. The maximal and minimal solutions are demonstrated. The analysis of eigenvalues and eigenfunctions is performed. It is discussed whether multiple solutions may exist for the inhomogeneous Sturm–Liouville boundary-value problem for differential equation with nonlocal conditions.

A study on existence and stability analysis of implicit k,Ψ$$ \left(k,\Psi \right) $$‐Hilfer fractional differential equations and application to RC$$ \mathcal{RC} $$ circuit model

In the present study, we establish the existence and uniqueness of solutions for nonlinear initial value problems and nonlocal boundary value problems associated with implicit fractional differential equations involving ‐Hilfer derivative operator. Furthermore, we explore the stability properties of these solutions in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized stability concepts by proving and applying generalized Gronwall inequality. Lastly, we present the fractional electric circuit model as an example to show the practical applicability of our main results.

Performance Analysis and Parallel Scalability of Numerical Methods for Fractional-in-Space Diffusion Problems with Adaptive Time Stepping

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (−Δ)α, 0<α<1, following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms.

Existence and Uniqueness Solutions of Multi-Term Delay Caputo Fractional Differential Equations

This study investigates a novel type of nonlocal boundary value problem with multipoint-integral boundaries and multi-term delay Caputo fractional differential equations (FDE). The provided problem is turned into an analogous fixed-point problem using fixed-point (F P) theory tools. Additionally, discussing about stability, in Ulam-Hyers-Rassias (UHR), Ulam-Hyers (UH), generalized Ulam-Hyers-Rassias (GUHR) and generalized Ulam-Hyers (GUH) stability, for finding the problem. Based on our obtained results we given some examples. As of our obtained results are very useful to multi-term caputo FDE related to hydrodynamics.

Existence theory for ψ -Caputo fractional differential equations

UDC 517.9 The target of this paper is to handle a nonlocal boundary-value problem for a specific kind of nonlinear fractional differential equations encapsuling a collective fractional derivative known as the ψ -Caputo fractional operator. The applied fractional operator generated by the kernel is of the following kind: k ( t , s ) = ψ ( t ) - ψ ( s ) . The existence of the solutions of the above-mentioned equations is tackled by using Mönch's fixed-point theorem combined with the technique of measures of noncompactness. In addition, we discuss the problem of stability within the scope of the Ulam–Hyers stability criteria for the main fractional system. Finally, an example is given to illustrate the viability of the reported results.

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments.

Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples.

Bitsadze-Samarsky type problems with double involution

In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green’s functions of the classical Dirichlet and Neumann boundary value problems, Green’s functions of the studied problems are constructed and integral representations of solutions to these problems are obtained.

Some Qualitative Results for Nonlocal Dynamic Boundary Value Problem of Thermistor Type

Some Qualitative Results for Nonlocal Dynamic Boundary Value Problem of Thermistor Type

On a Local and Nonlocal Second-Order Boundary Value Problem with In-Homogeneous Cauchy–Neumann Boundary Conditions—Applications in Engineering and Industry

A qualitative study for a second-order boundary value problem with local or nonlocal diffusion and a cubic nonlinear reaction term, endowed with in-homogeneous Cauchy–Neumann (Robin) boundary conditions, is addressed in the present paper. Provided that the initial data meet appropriate regularity conditions, the existence of solutions to the nonlocal problem is given at the beginning in a function space suitably chosen. Next, under certain assumptions on the known data, we prove the well posedness (the existence, a priori estimates, regularity, uniqueness) of the classical solution to the local problem. At the end, we present a particularization of the local and nonlocal problems, with applications for image processing (reconstruction, segmentation, etc.). Some conclusions are given, as well as new directions to extend the results and methods presented in this paper.

Non-local second-order boundary value problems with exponential nonlinearity

In this article, the Numerical treatment (Impilict-Explicite) was applied, for solving problems- related to the nonlocal second-order boundary value problem with exponential nonlinearity. The finite difference method of treatment of the BVP is used as implicit technique within the smallest available subdomain, . A reduced system of nonlinear algebraic equations corresponding to standard BVP is generated within the smallest domain, . A reduction of the computational work is introduced through reducing the size of the nonlinear algebraic system. A marching explicit approach is reemployed to generate the solution in , the complement of the interval with respect to the overall domain. Simple numerical experiments confirming the applicability of the treatment are introduced.

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

A new class of nonlocal multipoint boundary value problems involving a dual hybrid system of nonlinear Riemann–Liouville-type [Formula: see text]-fractional differential equations is studied in this paper. Existence and uniqueness results for the given problem are derived by applying the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle. Examples are presented for illustrating the obtained results. The work established in this paper is a useful contribution to the existing literature on [Formula: see text]-fractional differential equations. Some interesting special cases are also discussed.

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 a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green’s function on the real line. This free space Green’s function corresponds to the nonlocal boundary value problem proposed earlier by Saito [Appl. Comput. Harmon. Anal. 25, 68–97 (2008)]. We prove these perturbations to be polynomials of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

GRID-STEP OPTIMIZATION OF THE PARAMETERS OF THERMAL IMPACT ON THE TECHNICAL SYSTEM

The article solves applied problems of optimizing the parameters of technical systems under the influence of concentrated, discrete sources of thermal load. To improve the accuracy and speed of solving applied optimization problems, the authors propose a generalized procedure for finding the optimal thermal load parameters. This procedure consists of well-known computational methods for solving boundary value problems and ensuring the optimization of the objective function and its parameters. To calculate the values of the temperature field, it is necessary to solve a nonlocal boundary value problem of a system of nonlinear differential heat transfer equations. In order to find the conditions for the correctness of boundary value problems, the authors used the theory of pseudo-differential operators in the space of generalized slow power functions. This made it possible to guarantee the correctness of computational and applied optimization mathematical models. The procedure for grid-step optimization of the parameters of thermal action on a multilayer material proposed by the authors is based on a grid approach with discretization of the main parameters of thermal load and grinding of a large uniform grid in the vicinity of the selected node of the large grid. The directed search for local extremes of the mesh model is carried out by the nodes of the crushed mesh. The search for local extrema is performed by selecting the next node of a large grid. This approach will increase the accuracy of solving applied optimization problems by allowing further mesh refinement and refinement of the values of local extremes, and, consequently, refinement of the values of the global extremum of the objective function. The optimization of the thermal impact parameters was carried out according to the criterion of reducing damage to the test material. This makes it possible to increase the accuracy of the technological process of thermal action on a multilayer material.

NONLOCAL BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION ON THE HALF-LINE

This paper aims to investigate a class for nonlocal fractional boundary value problem on an infinite interval due to its importance in provide a powerful tool for mathematical modeling of complex phenomena in science. New existence results are acquired for the given problem by using the Krasnosel’skii’s fixed point theorem. Moreover, sufficient conditions are obtained as well as a modified compactness criterion that guarantees the existence of at least one solution. In addition, an illustrative example is given in the final part of the paper.

Bitsadze–Samarsky Type Nonlocal Boundary Value Problem for a Second Kind Mixed Equation with a Conjugation Condition of the Frankl Type

Bitsadze–Samarsky Type Nonlocal Boundary Value Problem for a Second Kind Mixed Equation with a Conjugation Condition of the Frankl Type

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.