Nonlocal Diffusion Equation Research Articles

Error estimates for Galerkin finite element approximations of time-fractional nonlocal diffusion equation

This paper is concerned to study the well-posedness, the Mittag–Leffler stability of solutions of time-fractional nonlocal reaction–diffusion equation in bounded domain We use the Faedo–Galerkin approximation method with initial data in to show a solution in Further, we construct the suitable Lyapunov function to ensure that a solution of the proposed model is the Mittag–Leffler stable. Furthermore, we fully discretize the Galerkin finite element method for the proposed time-fractional model in two-space dimension. Here, time-fractional derivative is given in Caputo's sense and discretized using approximation scheme. Error analysis of the proposed numerical method is performed and error bounds are obtained for the error measured in norm. All the theoretical results are validated with several constructive numerical examples.

Reconstruction of Timewise Term for the Nonlocal Diffusion Equation from an Additional Condition

The inverse problem of reconstructing the timewise term along with the temperature in the nonlocal diffusion equation with initial and boundary conditions and additional measurement (an integral and partial heat flux) is, for the first time, numerically investigated. This inverse problem appears extensively in the modeling of various phenomena in engineering and physics. The inverse problem considered in this paper has a unique solution. A Crank–Nicolson FDM is applied as a direct solver. The resulting nonlinear problem is solved using the MATLAB subroutine to minimize the objective functional. The Tikhonov regularization method is applied where necessary. A pair of examples, with smooth and non-smooth continuous timewise term, are discussed to assess the accuracy and stability of the numerical solutions.

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

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.

Finite element error analysis of a time-fractional nonlocal diffusion equation with the Dirichlet energy

A time-fractional diffusion equation involving the Dirichlet energy is considered with nonlocal diffusion operator in the space which has dimension d∈{2,3} and the Caputo sense fractional derivative in time. Further, nonlocal term in diffusion operator is of Kirchhoff type. We discretize the space using the Galerkin finite elements and time using the finite difference scheme on a uniform mesh. First, we prove the existence and uniqueness of a fully discrete numerical solution of the problem using the Brouwer fixed point theorem. Then, we give a priori bounds and convergence estimates in L2 and L∞ norms for fully-discrete problem. A more delicate analysis in the error provides the second order convergence for the proposed scheme. Numerical results are provided to validate the theoretical analysis.

Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity

This paper mainly focuses on the entire solutions of nonlocal dispersal equations with bistable nonlinearity. Under certain assumptions of wave speed, firstly constructing appropriate super- and sub-solutions and applying corresponding comparison principle, we established the existence and related properties of entire solutions formed by the collision of three and four traveling wave solutions. Then by introducing the definition of terminated sequence, it is proved that there has no entire solutions formed by $ k $ traveling wave solutions that collide with each other as long as $ k\geq5 $. Finally, based on the classical weighted energy approach, we obtain the global exponentially stability of the entire solutions in some weighted space.

Fourier Spectral Methods for Nonlocal Models

Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in Alali and Albin (Appl Anal :1–21, 2019). It is demonstrated that the Fourier multipliers, and the eigenvalues in particular, can be computed accurately and efficiently. This is achieved through utilizing the hypergeometric representation of the Fourier multipliers in which their computation in n dimensions reduces to the computation of a 1D smooth function given in terms of 2F3. We use this representation to develop spectral techniques to solve periodic nonlocal time-dependent problems. For linear problems, such as the nonlocal diffusion and nonlocal wave equations, we use the diagonalizability of the nonlocal operators to produce a semi-analytic approach. For nonlinear problems, we present a pseudo-spectral method and apply it to solve a Brusselator model with nonlocal diffusion. Accuracy and efficiency of the spectral solvers are discussed.

Anisotropic nonlocal diffusion equations with singular forcing

We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very little regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.

Blow-up of solutions for nonlocal diffusion equations with a weighted gradient reaction

Blow-up of solutions for nonlocal diffusion equations with a weighted gradient reaction

On the unique continuation of solutions to non-local non-linear dispersive equations

We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if are two suitable solutions of the equation defined in such that for some non-empty open set for then for any The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann, and some extensions obtained here.

Mathematical modelling of non-local spore dispersion of wind-borne pathogens causing fungal diseases

Theoretical description of epidemics of plant diseases is an invaluable resource for their efficient management. Here we propose a mathematical model for describing the dispersal by wind of fungal pathogens in plant populations. The dispersal of pathogen spores was modelled using a non-local diffusion equation which took into account variations in wind velocity components and contained a threshold in the convolution kernel defining the non-local diffusion term. The model was analyzed and the epidemic levels and patterns of the plant disease were derived, based upon defined assumptions of the time and space variables (i.e., represented by continuous parameters), and the host population (i.e., fixed population size). Numerical applications were then performed using reported characteristic values for wheat leaf rust, stripe rust and stem rust.

Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator [Formula: see text] and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalized principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter [Formula: see text]. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter [Formula: see text] is in a different range. In particular, for the case [Formula: see text], we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition.

Persistence and extinction of nonlocal dispersal evolution equations in moving habitats

This paper is devoted to the study of persistence and extinction of a species modeled by nonlocal dispersal evolution equations in moving habitats with moving speed c. It is shown that the species becomes extinct if the moving speed c is larger than the so called spreading speed c∗, where c∗ is determined by the maximum linearized growth rate function. If the moving speed c is smaller than c∗, it is shown that the persistence of the species depends on the patch size of the habitat, namely, the species persists if the patch size is greater than some number L∗ and in this case, there is a traveling wave solution with speed c, and it becomes extinct if the patch size is smaller than L∗.

Nonlocal diffusion equations with dynamical boundary conditions

In this paper we study nonlocal problems that are analogous to the local ones given by the Laplacian or the p−Laplacian with dynamical boundary conditions. We deal both with smooth and with singular kernels and show existence and uniqueness of solutions and study their asymptotic behaviour as t goes to infinity.

Stability and Error Analysis for a Second-Order Fast Approximation of the Local and Nonlocal Diffusion Equations on the Real Line

The stability and error analysis of a second-order fast approximation are considered for the one-dimensional local and nonlocal diffusion equations in the unbounded spatial domain. We first use the conventional central difference scheme to discretize the local second-order spatial derivative operator and use an asymptotically compatible difference scheme to discretize the spatial nonlocal diffusion operator, and apply second-order backward differentiation formula (BDF2) to approximate the temporal derivative to achieve a fully discrete infinity system. To solve the resulting fully discrete systems, we develop a unified framework that is applicable to the discretization of both local and nonlocal problems. A key ingredient is to derive Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs). To do so, we apply the $z$-transform and solve an exterior problem using an iteration technique to derive a Dirichlet-to-Dirichlet (DtD)-type mapping as exact ABCs. After that, we use the Green formula to reformulate the DtD-type mapping equivalently as the DtN-type mapping. The resulting DtN-type mapping allows us to reduce the infinity discrete system into a finite discrete system in a truncated computational domain of interest, and also make it possible to present the stability and convergence analysis of the reduced problem under some open but reasonable assumptions. To efficiently implement the exact ABCs, we further develop a fast convolution algorithm based on approximation of the contour integral induced by the inverse $z$-transform. The stability and error analysis of the reduced finite discrete system based on the fast algorithm for exact ABCs are also established, and numerical examples are provided to demonstrate the effectiveness of our proposed approach.

Traveling front of polyhedral shape for a nonlocal delayed diffusion equation

This paper is concerned with the existence and stability of traveling fronts with convex polyhedral shape for nonlocal delay diffusion equations. By using the existence and stability results of V-form fronts and pyramidal traveling fronts, we first show that there exists a traveling front V(x, y, z) with polyhedral shape of nonlocal delay diffusion equation associated with z = h(x, y). Moreover, the asymptotic stability and other qualitative properties of such traveling front V(x, y, z) are also established.

Self-Similar Behavior of a Nonlocal Diffusion Equation with Time Delay

We study the asymptotic behavior of solutions of a class of linear nonlocal measure valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a...

A spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions

The nonlocal diffusion model describes the diffusion process of solutes in complex media properly, while the classical theory of partial differential equations can not provide an appropriate description. The purpose of this paper is to provide illustrations from both theoretical and numerical perspectives of the computation of nonlocal diffusion models by Legendre collocation methods. Compared to local numerical methods, Legendre collocation methods can achieve a fixed accuracy with much fewer unknowns whenever the computational domain is regular and the solutions are sufficiently smooth. This paper is a groundwork towards efficient high order methods including spectral and spectral element methods for nonlocal diffusion equations with volume constrained boundary conditions.

New nonlocal forward model for diffuse optical tomography.

The forward model in diffuse optical tomography (DOT) describes how light propagates through a turbid medium. It is often approximated by a diffusion equation (DE) that is numerically discretized by the classical finite element method (FEM). We propose a nonlocal diffusion equation (NDE) as a new forward model for DOT, the discretization of which is carried out with an efficient graph-based numerical method (GNM). To quantitatively evaluate the new forward model, we first conduct experiments on a homogeneous slab, where the numerical accuracy of both NDE and DE is compared against the existing analytical solution. We further evaluate NDE by comparing its image reconstruction performance (inverse problem) to that of DE. Our experiments show that NDE is quantitatively comparable to DE and is up to 64% faster due to the efficient graph-based representation that can be implemented identically for geometries in different dimensions. The proposed discretization method can be easily applied to other imaging techniques like diffuse correlation spectroscopy which are normally modeled by the diffusion equation.

On the parabolic Harnack inequality for non-local diffusion equations

We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $$d\ge \beta $$ , where $$\beta \in (0,2]$$ is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times $$t>0$$ in dimensions $$d\ge \beta $$ . This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $$u_0 \in L^q_{loc}$$ for q larger than the critical value $$\tfrac{d}{\beta }$$ of the elliptic operator $$(-\Delta )^{\beta /2}$$ , a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than $$d=1$$ provided $$\beta >1$$ , since we prove that the local Harnack inequality holds if $$d<\beta $$ .

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

It has been observed in many numerical simulations, experiments and from various theoretical treatments that heat transport in one-dimensional systems of interacting particles cannot be described by the phenomenological Fourier's law. The picture that has emerged from studies over the last few years is that Fourier's law gets replaced by a spatially non-local linear equation wherein the current at a point gets contributions from the temperature gradients in other parts of the system. Correspondingly the usual heat diffusion equation gets replaced by a non-local fractional-type diffusion equation. In this review, we describe the various theoretical approaches which lead to this framework and also discuss recent progress on this problem.