Nonlocal Differential Operators Research Articles

Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

Poisson brackets between conserved quantities are a fundamental tool in the theory of integrable systems. The subclass of weakly nonlocal Poisson brackets occurs in many significant integrable systems. Proving that a weakly nonlocal differential operator defines a Poisson bracket can be challenging. We propose a computational approach to this problem through the identification of such operators with superfunctions on supermanifolds.

Trace formula for nonlocal differential operators

The problem of differential operators with nonlocal boundary condition appears, e.g., in scattering theory, diffusion processes and the other applicable fields. In this paper, we consider a class of differential operators with nonlocal boundary condition, and give its trace formula of Gelfand-Levitan type.

Inverse nodal problem for nonlocal differential operators

Inverse nodal problem consists in constructing operators from the given zeros of their eigenfunctions. The problem of differential operators with nonlocal boundary condition appears, e.g., in scattering theory, diffusion processes and the other applicable fields. In this paper, we consider a class of differential operators with nonlocal boundary condition, and show that the potential function can be determined by nodal data.

An efficient nonlocal variational method with application to underwater image restoration

Light is absorbed and scattered when it travels though water, which causes the captured underwater images often suffering from blurring, low contrast and color degradation. To overcome these problems, we propose a novel variational model based on nonlocal differential operators, in which the underwater image formation model is successfully integrated into the variational framework. The underwater dark channel prior (UDCP) and quad-tree subdivision methods are applied to the construction of underwater image formation model to estimate the transmission map and the global background light. Furthermore, we employ a fast algorithm based on the alternating direction method of multipliers (ADMM) to speed up the solving procedure. To reproduce color saturation, we perform a Gamma correction operation on the recovered image. Both the real underwater image application test and the simulation experiment demonstrate that the proposed underwater nonlocal total variational (UNLTV) approach achieves superb performance on dehazing, denoising, and improving the visibility of underwater images. In addition, six state-of-the-art algorithms are compared under different challenging scenes to evaluate their effectiveness and robustness. Extensive qualitative and quantitative experimental comparisons further validate the superiority of the proposed method.

Modeling and simulation of ice–water interactions by coupling peridynamics with updated Lagrangian particle hydrodynamics

In this work, a coupling method between the bond-based peridynamics model for solids and the updated Lagrangian particle hydrodynamics (ULPH) model of fluids is developed to simulate interaction between ice and seawater. The ULPH (Tu and Li in J Comput Phys 348:495–513, 2017) is the latest development of using non-local differential operators to model Navier–Stokes equations of Newtonian fluids, which has the advantages to preserve the consistency with partial differential equations and second-order accuracy in spatial gradient interpolation. On the other hand, we model the ice-like materials by using the bond-based peridynamics, which formulates equations of motions in terms of integro-differential equations instead of partial differential equations. For demonstration purpose, the numerical solution of a rigid ball impacting an ice plate floating in calm water is presented. The results obtained have shown the versatility as well as the accuracy of the method.

Analysis of 3D IS-LM macroeconomic system model within the scope of fractional calculus

A mathematical model providing an asymptotic description of macro-economic system is considered in this work. The system in general deals with performance, behaviour, decision-making of an economy as a whole and also the structure. Due to the complexities of this system, a more complex mathematical model is requested. In this work, we considered the extension of the model using some non-local differential operators and the stochastic approach where the given parameters are converted to normal distributions. We have presented the conditions of existence of uniquely exact solutions of the system using the fixed-point theorem approach. Each model is solved numerical via a newly introduced modified Adams-Bashforth for fractional differential equations. We presented numerical simulations for different values of fractional order. The models with the Atangana-Baleanu and Caputo differential operators provided us with new attractors.

Wellposedness and regularity of a nonlinear variable-order fractional wave equation

We prove the wellposedness of a nonlinear variable-order fractional wave differential equation that describes the vibration of a single particle attached to a viscoelastic material. We then prove that the high-order regularity of its solution depends on the behavior of the variable orders α(t) and β(t) of the fractional damping terms at t=0. In particular, we prove that the solution to the model recovers full regularity when the fractional damping terms exhibit (the physically relevant) elastic restoration force and integer-order damping at the initial time t=0, which, thus, eliminates the incompatibility between the nonlocal differential operators and the locality of classical initial condition.

A new approach to capture heterogeneity in groundwater problem: An illustration with an Earth equation

One of the major problem faced in modeling groundwater flow problems is perhaps how to capture heterogeneity of the geological formation within which the flow takes place. In this paper, we suggested applied a newly established approach to model real world problems that combines the concept of stochastic modeling in which parameters inputs are converted into distributions and the time differential operator is replaced by non-local differential operators. We illustrated this method with the Earth equation of groundwater recharge. For each case, we provided numerical and exact solution using the newly established numerical scheme and Laplace transform. We presented some numerical simulations. The numerical graphical representations let no doubt to think that this approach is the future way of modeling complex problems.

On a nonlocal problem involving a nonstandard nonhomogeneous differential operator

In this article, we are concerned with some class of nonlocal problems involving a nonstandard nonlocal and nonhomogeneous differential operator settled in Musielak–Orlicz–Sobolev spaces. We apply variational approach and look for nontrivial solutions, that is, local minimizers of the corresponding energy functional.

Traveling fronts for lattice neural field equations

We show existence and uniqueness of traveling front solutions to a class of neural field equations set on a lattice with infinite range interactions in the regime where the kinetics of each individual neuron is of bistable type. The existence proof relies on a regularization of the traveling wave problem allowing us to use well-known existence results for traveling front solutions of continuous neural field equations. We then show that the traveling front solutions which have nonzero wave speed are unique (up to translation) by constructing appropriate sub and super solutions. The spectral properties of the traveling fronts are also investigated via a careful study of the linear operator around a traveling front in co-moving frame where we crucially use Fredholm properties of nonlocal differential operators previously obtained by the author in an earlier work. For the spectral analysis, we need to impose an extra exponential localization condition on the interactions.

Efficient L1-based nonlocal total variational model of Retinex for image restoration

Characteristics of an image, such as smoothness, edge, and texture, can be better preserved using the nonlocal differential operator in image processing. We establish an L1-based nonlocal total variational (NLTVL1) model based on Retinex theory that can be solved by a fast computational algorithm via the alternating direction method of multipliers. Experiential results demonstrate that our NLTVL1 method has a good performance on enhancing contrast, eliminating the influence of nonuniform illumination, and suppressing noise. Furthermore, compared with previous works, including traditional Retinex methods and variational Retinex methods, our proposed approach achieves superior performance on edge and texture preservation and needs fewer iterations on recovering the reflectance image, which is illustrated by examples and statistics.

Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solution and analysis of its density

In this paper we study a fractional stochastic heat equation on Rd(d ≥ 1) with additive noise ∂∂tu(t,x)=dδ_α_u(t,x)+b(u(t,x))+W˙H(t,x) where dδ_α_ is a nonlocal fractional differential operator and W˙H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d=1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.

The Dirichlet problem for stable-like operators and related probabilistic representations

ABSTRACTWe study stochastic differential equations with jumps with no diffusion part, governed by a large class of stable-like operators, which may contain a drift term. For this class of operators, we establish the regularity of solutions to the Dirichlet problem up to the boundary as well as the usual stochastic characterization of these solutions. We also establish key connections between the recurrence properties of the jump process and the associated nonlocal partial differential operator. Provided that the process is positive (Harris) recurrent, we also show that the mean hitting time of a ball is a viscosity solution of an exterior Dirichlet problem.

The total and updated lagrangian formulations of state-based peridynamics

The peridynamics theory is a reformulation of nonlocal continuum mechanics that incorporates material particle interactions at finite distances into the equation of motion. State-based peridynamics is an extension of the original bond-based peridynamics theory wherein the response of an individual particle depends collectively on its interaction with neighboring particles through the concept of state variables. In this paper, the more recent non-ordinary state-based Peridynamics formulations of both the total (referential) Lagrangian approach as well as the updated (spatial) Lagrangian approach are formulated. In doing so, relations of the state variables are defined through various nonlocal differential operators in both material and spatial configurations in the context of finite deformation. Moreover, these nonlocal differential operators are mathematically and numerically shown to converge to the local differential operators, and they are applied to derive new force states and deformation gradients.

FRACTIONAL MODELS IN SPACE FOR DIFFUSIVE PROCESSES IN HETEROGENEOUS MEDIA WITH APPLICATIONS IN CELL MOTILITY AND ELECTRICAL SIGNAL PROPAGATION

This work addresses fundamental issues in the mathematical modelling of the diffusive motion of particles in biological and physiological settings. New mathematical results are proved and implemented in computer models for the colonisation of the embryonic gut by neural cells and the propagation of electrical waves in the heart, offering new insights into the relationships between structure and function. In particular, the thesis focuses on the use of non-local differential operators of non-integer order to capture the main features of diffusion processes occurring in complex spatial structures characterised by high levels of heterogeneity.

Study on A Hierarchical Nonlocal Image Segmentation Method

In this paper, a hierarchical nonlocal method for image segmentation is proposed. The method is mainly consisting of two stages. In the first stage, a nonlocal segmentation model based on nonlocal differential operators is used to find a smooth solution. Once the solution is obtained, the segmentation is done by thresholding the solution into different phases in the second stage. The K-means method is used to determine the thresholds. One can obtain any K-phase segmentation (

Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations

In this work we consider the problems $$ \left\{\begin{array}{rcll} \mathcal{L \,} u&=&f &\hbox{ in } \Omega,\\ u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega, \end{array} \right. $$ and $$ \left\{\begin{array}{rcll} u_t +\mathcal{L \,} u&=&f &\hbox{ in } Q_{T}\equiv\Omega\times (0, T),\\ u (x,t) &=&0 &\hbox{ in } \big(\mathbb{R}^N\setminus\Omega\big) \times (0, T),\\ u(x,0)&=&0 &\hbox{ in } \Omega, \end{array} \right. $$ where $\mathcal{L \,}$ is a nonlocal differential operator and $\Omega$ is a bounded domain in $\mathbb{R}^N$, with Lipschitz boundary. &nbsp The main goal of this work is to study existence, uniqueness and summability of the solution $u$ with respect to the summability of the datum $f$. In the process we establish an $L^p$-theory, for $p \geq 1$, associated to these problems and we prove some useful inequalities for the applications.

Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems

The 6=3×2 huge Lie algebra Ξ of all local and non-local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket scheme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter (GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consistent way a wide class of integrable systems. Other algebraic properties are also presented.

Nonlocal TV-L1 Inpainting Model and its Augmented Lagrangian Algorithm

Nonlocal differential operators have been extensively applied to variational models for image restoration due to its texture-preserving capability. In this paper, we propose a nonlocal TV (total variation)-L1 model for texture image inpainting, which, technically, combines nonlocal operators for regularization term and L1 norm for data term. The former is used to regularize texture and the latter to preserve contrast of images. In addition, we develop augmented Lagrangian algorithm for proposed model by introducing nonlocal auxiliary variable and Lagrangian multiplier. Finally, extensive experiments on synthetic and real texture images are presented to validate the effectiveness and efficiency of our proposed model and algorithm.

Fredholm properties of nonlocal differential operators via spectral flow

We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear and a linear setting. We first prove the existence of small viscous shock waves in nonlocal conservation laws with small spatially localized source terms. We also show how our results can be used to study edge bifurcations in eigenvalue problems using Lyapunov-Schmidt reduction instead of a Gap Lemma.