Functional a Posteriori Estimates for the Fractional Laplacian Problem

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

Some bifurcation results for fractional Laplacian problems

PurposeThe purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.Design/methodology/approachThe authors first transform the given nonlocal boundary value problems of first‐ and second‐order differential equations into local boundary value problems of second‐ and third‐order differential equations, respectively. Then a modified Adomian decomposition method is applied, which permits convenient resolution of these equations.FindingsThe new technique, as presented in this paper in extending the applicability of the Adomian decomposition method, has been shown to be very efficient for solving nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.Originality/valueThe paper presents a new solution algorithm for the nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$ Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. Also we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G)$. In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data uD by functions uD,h in the trace space of a finite element space on ΓD. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of uD,h, namely the nodal interpolation and the orthogonal projection in L2(ΓD) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H1 and L2 a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.

Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

Tempered fractional Sobolev spaces

Our aim in this paper is to analyze the existence of solutions to a nonlocal elliptic problem involving the fractional Laplacian operator. These operators are utilized to solve an equation defined within a bounded domain in . The operator is a fractional Laplacian , where with the condition and is a non-negative function almost everywhere with respect to the variable , and are real numbers. The article establishes two existence theorems for weak solutions using Tychonoff and Schauder fixed-point theorems. These theorems are formulated based on different hypotheses regarding the parameters and .

In this paper we introduce a method to define fractional operators using mean value operators. In particular we discuss a geometric approach in order to construct fractional operators. As a byproduct we define fractional linear operators in Carnot groups, moreover we adapt our technique to define some nonlinear fractional operators associated with the $p-$Laplace operators in Carnot groups.

In this paper, we consider a nonlocal problem. It is used in the rectangular area. For this problem, a difference problem is constructed, based on a rectangular grid. The error of theapproximatesolutionsfor a nonlocal problemis estimated. Many applied problems, in particular heat conductivity [1], [2], fluid mechanics [3], the theory of elasticity and shells [4], etc. are reduced to solving nonlocal boundary value problems for elliptical type equations. When solving nonlocal boundaryvalueproblems, nonlocal boundaryconditions complicate the justification of finite difference schemes due to the complexity of matrices obtained from systemsof equation.This complexity is especially evident for numerical methods, fornon-linear equations[5]. Here we consider a nonlocal boundary value problem for a quasi-linear equation. To solve the problem numerically we use the finite difference method. The error estimate of the approximate solution of the nonlocal problem is found.

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator where and the fractional derivatives , , are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operator in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving (k1,ψ1)-Hilfer and (k2,ψ2)-Caputo fractional derivative operators, and (k2,ψ2)- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the (k1,ψ1)-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ψ1 and the parameter β. Also the (k2,ψ2)-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ψ2-Riemann–Liouville, k2-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and L1-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study.

Nonlocal Hénon type problem with nonlinearities involving slightly subcritical growth