Nonlocal Diffusion Research Articles

2D Problem of a Nonlocal Thermoelastic Diffusion Solid with Gravity via Three Theories

PurposeThe impact of gravity on nonlocal thermoelastic diffusion solid is discussed in this work.Methods The Green-Naghdi theory (G-N II), the Lord-Shulman theory, and the three-phase-lag model all explore the issue. The governing equations are solved using the normal mode technique to get the analytic forms of the displacements, temperatures, force stress tensors, and mass concentration. Using appropriate boundary conditions, the physical fields are calculated and the numerical computations have been carried out with the help of MATLAB programming.Results In the physical domain, numerical results for the field quantities are provided and graphically displayed in both the absence and presence of gravity and the nonlocal parameter.ConclusionPhysical variables are affected by nonlocal thermoelasticity as well as the gravity field.

Decay rates of convergence for Fokker-Planck equations with confining drift

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.

Fine patterns for a nonlocal periodic-parabolic equation with spatial degeneracy

In this paper, we investigate several properties of the positive solution to the nonlocal periodic-parabolic problem{ut=d[∫ΩJ(x−y)u(y,t)dy−u(x,t)]+λu−b(x)q(t)up in Ω×[0,T],u(x,0)=u(x,T) in Ω, where Ω is a bounded smooth domain in RN (N≥1), λ>0 is a parameter, d>0 and p>1 are constants, q(t) is a positive and T-periodic continuous function, b(x) is a nonnegative continuous function, and is strictly positive only outside a domain Ω¯0⊂Ω. So the equation has a degeneracy over Ω0. In such a case, it is known that there is a unique positive solution uλ if and only if λ is between the principle eigenvalues λp(Ω) and λp(Ω0) of the associated nonlocal diffusion operator (over Ω and Ω0, respectively). We show that uλ is stable under perturbations of λ and b(x) in the equation, and when λ increases to λp(Ω0), uλ converges to ∞ uniformly over Ω¯×[0,T] but a clear pattern is exhibited by uλ/‖uλ‖∞:uλ(x,t)‖uλ‖∞→{0 for (x,t)∈(Ω¯∖Ω¯0)×[0,T],ϕ(x) for (x,t)∈Ω¯0×[0,T], where ϕ(x) is the normalised positive eigenfunction associated to λp(Ω0). The proof relies on a general regularity result for the solutions of these types of nonlocal equations, which is of independent interest.

A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations

We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp.

Crank-Nicolson finite element methods for nonlocal problems with p -Laplace-type operator

A theoretical analysis of a Crank-Nicolson Galerkin finite element method for a class of nonlinear nonlocal diffusion problems associated with p -Laplace-type operator is presented here. It is shown, by a rigorous analysis that the unconditionally optimal error estimates for the fully discrete scheme are established. The presence of the nonlocal term in the models destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, a new algorithm is proposed to avoid the full Jacobian matrix. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains

Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long distance in either the forward or backward direction along preferential channels during the transport. The classical advection–diffusion (ADE) model has been widely used to describe tracer transport; however, they often fall short in capturing the intricacies of nonlocal diffusion processes. The fractional operator has gained recognition among hydrologists due to its potential to capture distinct mechanisms of transport and storage for tracer particles exhibiting nonlocal dynamics. However, the hypersingularity of the fractional Laplacian operator presents considerable difficulties in its numerical approximation in bounded domains. This study focuses on the development and application of the fractional Laplacian-based model to characterize nonlocal tracer transport behavior, specifically considering both forward and backward diffusion processes on bounded domains. The Riesz fractional Laplacian provides a mathematical framework for describing tracer diffusion processes on a bounded domain, and a novel finite difference method (FDM) is introduced as an effective numerical solver for addressing the fractional Laplacian-based model. Applications reveal that the fractional Laplacian-based model can effectively capture the observed nonlocal tracer transport behavior in a heterogeneous system, and nonlocal tracer transport exhibits dynamic characteristics, influenced by the evolving heterogeneity of the media at various temporal scales.

A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

Non-local diffusion problems with Neumann type boundary values

In this paper, we study the non-local diffusion problemJ⁎u−u−ut=f(u) with non-local Neumann boundary condition. We first establish some integration by parts formulas similar to the Laplacian. Then the existence and uniqueness of the solution and the maximum principle are proved. We also investigate the properties of the Neumann condition for one equation and for a family of equations.

Spreading speeds of a nonlocal diffusion model with free boundaries in the time almost periodic media

In this paper, we mainly investigate the spreading dynamics of a nonlocal diffusion KPP model with free boundaries which is firstly explored in time almost periodic media. As the spreading occurs, the long-run dynamics are obtained. Especially, when the threshold condition for the kernel function is satisfied, applying the novel positive time almost periodic function, we accurately express the unique asymptotic spreading speed of the free boundary problem.

Data-driven modelling and dynamic analysis of the multistable energy harvester with non-Gaussian Lévy noise

In engineering, due to the complex structural characteristics of system and the non-Gaussian properties of random excitation, it is difficult to establish an accurate stochastic dynamic model for the strongly nonlinear multistable vibration energy harvester (VEH), especially for these driven by non-Gaussian Lévy noise. From the view of machine learning, a data-driven model identification method is devised to extract the non-Gaussian governing laws of the multistable VEH with the aid of the observed sample trajectory data. Based on the Nonlocal Kramers-Moyal formulas, the Lévy, drift and diffusion terms can be approximately expressed by the sample trajectories of the system. By implementing the least square method and the stepwise sparse regressor algorithm, the optimal drift and diffusion coefficients can be identified, and then the non-Gaussian stochastic differential equation of VEH is extracted. Two examples are utilized to verify the feasibility and effectiveness of the data-driven modelling method in VEH, which indicates that the identified results agree well with the original system. Finally, the stochastic dynamic behaviors induced by non-Gaussian Lévy noise are explored based on the data-driven penta-stable VEH. The proposed method can provide the theoretical guidance for the modelling and dynamics research of VEH in engineering.

A high-order and efficient numerical technique for the nonlocal neutron diffusion equation representing neutron transport in a nuclear reactor

In this paper, a high-order and efficient numerical technique is constructed to solve nonlocal neutron diffusion equation with delayed neutrons representing neutron transport in a nuclear reactor. The method is based on approximating the temporal derivative by L1-2 technique, in combination with a space discretization by using compact difference method. The convergence of this method are studied using energy analysis and Cholesky decomposition. The error convergence order is shown to be O(k3−2α+h4), where α is the order of fractional derivative, k and h represent the parameters for the time and space meshes, respectively. Further, two numerical experiments are presented to validate the sharpness of our theoretical error bounds.

Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion

This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined.

Stability and uniqueness of generalized traveling front for non-autonomous Fisher–KPP equations with nonlocal diffusion

This paper is concerned with the stability and uniqueness of generalized traveling front of non-autonomous Fisher–KPP equations with nonlocal diffusion. In such non-autonomous nonlocal dispersal equations, both the diffusion kernel and the reaction term generally depend on time. The existence of generalized traveling wave fronts of such equations has been studied in Ducrot and Jin (2022). By using comparison principle and part metric, we show the generalized traveling front is asymptotically stable under well-fitted perturbation and it is also unique.

Nlfem: a Flexible 2d FEM Python Code for Nonlocal Convection-Diffusion and Mechanics

In this work, we present the mathematical foundation of an assembly code for finite element approximations of nonlocal models with compactly supported, weakly singular kernels. We demonstrate the code on a nonlocal diffusion model in various configurations and on a two-dimensional bond-based peridynamics model. Further examples can be found in D’Elia et al. (Math Models Methods Appl Sci 31(08):1505–1567, 2021). The code nlfem is published under the GNU General Public License (for details, see, e.g., https://www.gnu.org/licenses/gpl-3.0.de.html) and can be freely downloaded at https://gitlab.uni-trier.de/pde-opt/nonlocal-models/nlfem.

Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a “projection” of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one.Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.

Asymptotical profiles of an age-structured foot-and-mouth disease with nonlocal diffusion on a spatially heterogeneous environment

Spatial heterogeneity deeply affects the habits of animals and transmission behaviors of foot-and-mouth disease. To consider the effects of heterogeneous habits on the evolution of the disease prevalence, an age-structured foot-and-mouth disease (FMD) model with nonlocal diffusion is proposed to describe the long distance movements of cattle. The next generation operator G is derived through establishing an extensively renewal equation and then the basic reproduction number R0 is the spectral radius of G. Moreover, the value of R0 serves as a threshold which distinguishes between the disappearance and an outbreak of FMD. The outbreak of FMD is the presentation of the endemic steady state by using the Krasnoselskii fixed point theorem. Finally, numerical simulations shed light on how the magnitudes of nonlocal diffusions affect the evolution of FMD spread.

Dynamics analysis of a nonlocal diffusion dengue model

Due to the unrestricted movement of humans over a wide area, it is important to understand how individuals move between non-adjacent locations in space. In this research, we introduce a nonlocal diffusion introduce for dengue, which is driven by integral operators. First, we use the semigroup theory and continuously Fréchet differentiable to demonstrate the existence, uniqueness, positivity and boundedness of the solution. Next, the global stability and uniform persistence of the system are proved by analyzing the eigenvalue problem of the nonlocal diffusion term. To achieve this, the Lyapunov function is derived and the comparison principle is applied. Finally, numerical simulations are carried out to validate the results of the theorem, and it is revealed that controlling the disease’s spread can be achieved by implementing measures to reduce the transmission of the virus through infected humans and mosquitoes.

Exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries

Accelerated propagation is a new phenomenon associated with nonlocal diffusion problems. In this paper, we determine the exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries, where the nonlocal diffusion operator is given by ∫RJ(x-y)u(t,y)dy-u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\displaystyle \\int _{\\mathbb {R}}J(x-y)u(t,y)dy-u(t,x)$$\\end{document}, and the kernel function J(x) behaves like a power function near infinity, namely lim|x|→∞J(x)|x|α=λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim _{|x|\\rightarrow \\infty } J(x)|x|^{\\alpha }=\\lambda >0$$\\end{document} for some α∈(1,2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2]$$\\end{document}. This is the precise range of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} where accelerated spreading can happen for such kernels. By constructing subtle upper and lower solutions, we prove that the location of the free boundaries x=h(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=h(t)$$\\end{document} and x=g(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=g(t)$$\\end{document} goes to infinity at exactly the following rates: limt→∞h(t)tlnt=limt→∞-g(t)tlnt=μλ,whenα=2,limt→∞h(t)t1/(α-1)=limt→∞-g(t)t1/(α-1)=22-α2-αμλ,whenα∈(1,2).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle \\lim _{t\\rightarrow \\infty }\\frac{h(t)}{t\\ln t}=\\lim _{t\\rightarrow \\infty }\\frac{-g (t)}{t\\ln t}=\\mu \\lambda ,&{} \\hbox { when } \\alpha =2,\\\\ \\displaystyle \\lim _{t\\rightarrow \\infty }\\frac{h(t)}{ t^{1/(\\alpha -1)}}= \\lim _{t\\rightarrow \\infty }\\frac{-g (t)}{ t^{1/(\\alpha -1)}}=\\frac{2^{2-\\alpha }}{2-\\alpha }\\mu \\lambda , &{} \\hbox { when } \\alpha \\in (1,2). \\end{array}\\right. } \\end{aligned}$$\\end{document}Here μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu >0$$\\end{document} is a given parameter in the free boundary condition. Accelerated propagation can also happen when lim|x|→∞J(x)|x|(ln|x|)β=λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim _{|x|\\rightarrow \\infty }J(x)|x|(\\ln |x|)^\\beta =\\lambda >0$$\\end{document} for some β>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >1$$\\end{document}. For this case, we prove that -g(t),h(t)=exp{[(2βμλβ-1)1/β+o(1)]t1/β}ast→∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -g (t), h(t)=\\exp \\Big \\{\\Big [\\Big (\\frac{2\\beta \\mu \\lambda }{\\beta -1}\\Big )^{1/\\beta }+o(1)\\Big ]t^{1/\\beta }\\Big \\} \\hbox { as } t\\rightarrow \\infty . \\end{aligned}$$\\end{document}These results considerably sharpen the corresponding ones in [20], and the techniques developed here open the door for obtaining similar precise results for other problems. A crucial technical point is that such precise conclusions on the propagation are achievable by finding the correct improvements on the form of the lower solutions used in [20], even though the precise long-time profile of the density function u(t, x) is still lacking.

Spreading Properties for Non-autonomous Fisher–KPP Equations with Non-local Diffusion

Spreading Properties for Non-autonomous Fisher–KPP Equations with Non-local Diffusion

The dissipative Generalized Hydrodynamic equations and their numerical solution

“Generalized Hydrodynamics” (GHD) stands for a model that describes one-dimensional integrable systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic type, where the main unknown, a statistical distribution function f(t,z,θ), lives in a phase space which is constituted by a one-dimensional position variable z, and a one-dimensional “kinetic” variable θ, actually a wave-vector, called “rapidity”. Two key features of GHD equations are first a non-local and nonlinear coupling in the advection term, and second an infinite set of conserved quantities, which prevent the system from thermalizing. To go beyond this, we consider the dissipative GHD equations, which are obtained by supplementing the right-hand side of the GHD equations with a non-local and nonlinear diffusion operator or a Boltzmann-type collision integral. In this paper, we deal with new high-order numerical methods to efficiently solve these kinetic equations. In particular, we devise novel backward semi-Lagrangian methods for solving the advective part (the so-called Vlasov equation) by using a high-order time-Taylor series expansion for the advection fields, whose successive time derivatives are obtained by a recursive procedure. This high-order temporal approximation of the advection fields is used to design new implicit/explicit Runge–Kutta semi-Lagrangian methods, which are compared to Adams–Moulton semi-Lagrangian schemes. For solving the source terms, constituted by the diffusion and collision operators, we use and compare different numerical methods of the literature.