Nonlocal Diffusion Equation Research Articles

Collocation method for one dimensional nonlocal diffusion equations

In this paper, the collocation method for solving one dimensional steady state and time dependent nonlocal diffusion equations is analyzed. The difficulty of applying collocation method to nonlocal diffusion equations comes from the singularity of the kernel. If the kernel is weakly singular, however, if the kernel is not integrable in Riemann sense. So that the Hadamard finite part integral is introduced to overcome this difficulty. For analysis and performance, a “balance” term is added to discretize the nonlocal operator. Numerical results validate the theorems.

Multi-type forced waves in nonlocal dispersal KPP equations with shifting habitats

The current paper is concerned with the forced waves to nonlocal dispersal KPP equations with shifting habitats. Without asking the sign of intrinsic growth rate at negative infinity, we derive the conditions for the existence of multi-type forced waves with different profiles corresponding to different shifting speeds. Our results show that (i) there exist not only monotone but also non-monotone forced waves, and (ii) even at the same wave speed as the shifting speed of habitats, more than one forced waves can be found.

Decay and symmetry of solitary waves

In this paper we consider the decay rate of solitary-wave solutions to some classes of non-linear and non-local dispersive equations, including for example the Whitham equation and a Whitham–Boussinesq system. The dispersive term is represented by a Fourier multiplier operator that has a real analytic symbol that either decays/grows, and we show that all supercritical/subcritical solitary-wave solutions decay exponentially, and moreover provide the exact decay rate, which in general will depend on the speed of the wave. We also prove that solitary waves have only one crest and are symmetric for some class of equations.

Effects of dispersal and spatial heterogeneity on nonlocal logistic equations

In the present paper, we are interested in the nonlocal dispersal logistic equations in heterogeneous environment. This leads to study the spectrum theory and asymptotic behavior of nonlocal dispersal problems. We establish the existence and uniqueness of positive stationary solutions. By employing continuous lemma and nonlocal estimates, we obtain the limiting behavior of positive solutions when the dispersal rate is small or large. We also analyze the effect of spatial heterogeneity on the long time behavior of evolution equations.

Asymptotic propagations of a nonlocal dispersal population model with shifting habitats

This paper is concerned with the asymptotic propagations for a nonlocal dispersal population model with shifting habitats. In particular, we verify that the invading speed of the species is determined by the speed c of the shifting habitat edge and the behaviours near infinity of the species’ growth rate which is nondecreasing along the positive spatial direction. In the case where the species declines near the negative infinity, we conclude that extinction occurs if c > c*(∞), while c < c*(∞), spreading happens with a leftward speed min{−c, c*(∞)} and a rightward speed c*(∞), where c*(∞) is the minimum KPP travelling wave speed associated with the species’ growth rate at the positive infinity. The same scenario will play out for the case where the species’ growth rate is zero at negative infinity. In the case where the species still grows near negative infinity, we show that the species always survives ‘by moving’ with the rightward spreading speed being either c*(∞) or c*(−∞) and the leftward spreading speed being one of c*(∞), c*(−∞) and −c, where c*(−∞) is the minimum KPP travelling wave speed corresponding to the growth rate at the negative infinity. Finally, we give some numeric simulations and discussions to present and explain the theoretical results. Our results indicate that there may exists a solution like a two-layer wave with the propagation speeds analytically determined for such type of nonlocal dispersal equations.

Coupling local and nonlocal diffusion equations for image denoising

Due to the strong ability of restoring textures and details in images, nonlocal equations have attracted an extensive interest for image denoising. However, the lack of regularity causes residual noise in the restored images. In this paper, we propose and study an evolution equation consisting of the weighted local and the weighted nonlocal p-Laplacian equations. The existence and uniqueness of solutions for the proposed equation are proven under the assumption that the weights vanish in sets of positive measure. We also show that solutions of the proposed equation converge to the solution of the usual p-Laplacian equation if the kernel is rescaled appropriately. Comparisons with local and nonlocal diffusion equations for removing Gaussian noise in images are presented.

Variational Bayesian inversion for the reaction coefficient in space-time nonlocal diffusion equations

In the paper, a variational Bayesian method is used to identify the reaction coefficient for space-time nonlocal diffusion equations using nonlocal averaged flux data. To show the posterior measure to be well-defined, we rigorously prove that the forward operator is continuous with respect to the unknown reaction field. Then, gradient-based prior information is proposed to explore oscillation features in the reaction coefficient. Moreover, the Bayesian inverse problem is shown to be well-posed in Hellinger distance. To accurately characterize the posterior density using uncorrelated samples, an efficient variational Bayesian method is used to estimate the reaction coefficient in the nonlocal models. A few numerical results are presented to illustrate the efficacy of the proposed approach and confirm some theoretic discoveries.

Propagation dynamics of an anisotropic nonlocal dispersal equation with delayed nonlocal response

This paper focuses on the critical wave speed and the traveling wave fronts for a general anisotropic nonlocal dispersal equation with delayed nonlocal response. Unlike most of the previous study, both of the dispersal kernel function and the nonlocal response function are asymmetric in such an equation. By analyzing the properties of eigenfunction, we first discuss the sign of the critical wave speed and the influence of the asymmetry of the two kernel functions on the critical wave speed. Then under the monostable condition, by using super- and subsolution and monotone iteration method, we obtain the existence of nondecreasing traveling wave solution for c>c∗ and a limiting argument for c=c∗. Moreover, we depict the asymptotic behavior of the traveling wave and its derivative at minus infinity.

Image denoising by nonlinear nonlocal diffusion equations

In this paper, we propose and study a nonlinear nonlocal diffusion equation of the Perona–Malik type for the removal of Gaussian noise in images. Based on the fixed-point theorem, we prove the unique solvability of the equation. We also show that solutions of the nonlocal equation converge to the solution of the local spatially regularized Perona–Malik equation if the kernel is rescaled appropriately. The new equation inherits the merit of the nonlocal method that restores details and textures of noisy images but avoids speckles and artifacts in homogeneous regions. Comparisons with other nonlocal methods for image denoising are presented.

Nonplanar traveling fronts for nonlocal dispersal equations with bistable nonlinearity

This paper is concerned with nonplanar traveling fronts for a class of nonlocal dispersal equations with bistable nonlinearity. A comparison principle is introduced and employed to establish the existence of traveling fronts with pyramidal shape in $\mathbb R^3$. Although the speed of the pyramidal traveling fronts is not unique, their global average speed is unique. This is possibly the first time that the nonplanar traveling waves for the nonlocal dispersal equations has been studied.

The effects of degeneracy on nonlocal dispersal logistic equations

This paper is concerned with the nonlocal dispersal logistic equation J∗u(x)−u(x)=−λu(x)+a(x)up in Ω̄,u(x)=0 in RN∖Ω̄,where Ω⊂RN(N≥1) is a bounded domain, λ and p>1 are constants, the dispersal kernel J and the coefficient a(x) are nonnegative. The work of García-Melián and Rossi (2009) reveals that the nonlocality makes a key change of positive solutions. In this paper, we investigate the sharp effects of degeneracy on the behavior of positive solutions. We find that the effect is different from the diffusive logistic equation.

Spatial propagation in nonlocal dispersal Fisher-KPP equations

In this paper we focus on three problems about the spreading speeds of nonlocal dispersal Fisher-KPP equations. First, we study the signs of spreading speeds and find that they are determined by the asymmetry level of the nonlocal dispersal and f′(0), where f is the reaction function. This indicates that asymmetric dispersal can influence the spatial dynamics in three aspects: it can determine the spatial propagation directions of solutions, influence the stability of equilibrium states, and affect the monotone property of solutions. Second, we give an improved proof of the spreading speed result by constructing new lower solutions and using the new “forward-backward spreading” method. Third, we investigate the relationship between spreading speed and exponentially decaying initial data. Our result demonstrates that when dispersal is symmetric, spreading speed decreases along with the increase of the exponential decay rate. In addition, the results on the signs of spreading speeds are applied to two special cases where we present more details on the influence of asymmetric dispersal.

Non-blowup at critical exponent for a semilinear nonlocal diffusion equation

In this work, we investigate the positive solutions of a semilinear nonlocal diffusion equation with power nonlinearity. We have discovered a new phenomenon that the equation at the critical exponent, under a certain condition on the kernel function, admits a global in time solution for a small initial datum. This is in sharp contrast to the corresponding heat equation and the nonlocal diffusion equation with a regular or fractional Laplacian kernel, where every (nontrivial) positive solution always blows up in finite time at the critical exponent.

Determination of the reaction coefficient in a time dependent nonlocal diffusion process

This paper investigates the inverse reaction coefficient problem for a time dependent nonlocal diffusion equation by utilizing the nonlocal flux measurement from an accessible part of region, which is a continuation and an extension of our recent work (Zheng and Ding 2020 Inverse Problems 36 035006). The uniqueness of inverse reaction coefficient problem is proved. The variational regularization method is proposed to overcome the ill-posedness of inverse problem, and an alternating iteration method is adopted to automatically select regularization parameters. Furthermore, we use Laplace approximation method to capture the statistics information of the solution. In particular, we prove the lower bound of confidence width is rigorously estimated to characterize the reliability of the method. Finally, numerical results indicate that the proposed method could yield an accurate estimate and efficient uncertainty quantification of the solution.

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.

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

In this work, we study the final value problem for a system of parabolic diffusion equations. In which, the final value functions are derived from a random model. This problem is severely ill‐posed in the sense of Hadamard. By nonparametric estimation and truncation methods, we offer a new regularized solution. We also investigate an estimate of the error and a convergence rate between a mild solution and its regularized solutions. Finally, some numerical experiments are constructed to confirm the efficiency of the proposed method.

Determination of Time-Dependent Sources and Parameters of Nonlocal Diffusion and Wave Equations from Final Data

Abstract Two inverse problems with final overdetermination for diffusion and wave equations containing the Caputo fractional time derivative and a fractional Laplacian of distributed order are considered. They are: 1) the problem to reconstruct a time-dependent source term; 2) the problem to recover simultaneously the source term, the order of the time derivative and the fractional Laplacian. Uniqueness of solutions of these problems is proved. Sufficient conditions for the uniqueness are stricter for the 2nd problem than for the 1st problem.

Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity

<p style='text-indent:20px;'>This paper is concerned with the nonlocal dispersal equations with inhomogeneous bistable nonlinearity in one dimension. The varying nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. We establish the existence of a unique entire solution connecting two traveling wave solutions pertaining to the different nonlinearities. In particular, we use a "squeezing" technique to show that the traveling wave of the equation with one nonlinearity approaching from infinity, after going through the transition region, converges to the other traveling wave prescribed by the nonlinearity on the other side. Furthermore, we also prove that such an entire solution is Lyapunov stable.</p>

Two-dimensional nonlinear time fractional reaction\u2013diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media

In the present article, an efficient operational matrix based on the famous Laguerre polynomials is applied for the numerical solution of two-dimensional non-linear time fractional order reaction–diffusion equation. An operational matrix is constructed for fractional order differentiation and this operational matrix converts our proposed model into a system of non-linear algebraic equations through collocation which can be solved by using the Newton Iteration method. Assuming the surface layers are thermodynamically variant under some specified conditions, many insights and properties are deduced e.g., nonlocal diffusion equations and mass conservation of the binary species which are relevant to many engineering and physical problems. The salient features of present manuscript are finding the convergence analysis of the proposed scheme and also the validation and the exhibitions of effectiveness of the method using the order of convergence through the error analysis between the numerical solutions applying the proposed method and the analytical results for two existing problems. The prominent feature of the present article is the graphical presentations of the effect of reaction term on the behavior of solute profile of the considered model for different particular cases.

Lower bounds for some nonlocal dispersal equations

Let u(x,t) be the unique solution of the nonlocal dispersal equation ut=J⁎u−u in the whole Rn with initial value u0(x). In this paper, we establish the lower bounds for positive solution. Roughly speaking, we obtainu(x0,t)≥(4παt)−n/2e−(1+Lα)te−r24αt∫B(x0,r)u0(y)dy.