Nonlocal Differential Operators Research Articles

General particle dynamics: On the correlation of nonlocal and local theories for large deformation problems

This paper presents a detailed investigation on the General Particle Dynamics (GPD) method and its implications for addressing challenges in large deformation mechanics and fracture modeling. The GPD method integrates nonlocal interactions and nonlocal differential operators to overcome limitations of classical continuum mechanics in capturing long-range interactions and material behaviors under extreme deformation conditions. Through a systematic examination of the physical and mathematical foundations of GPD, including its governing equations and constitutive relations, we elucidate the robust and applicability of the proposed method in simulating complex deformation phenomena. Furthermore, the relationships among GPD, conventional mechanical theories (Smoothed Particle Hydrodynamics), nonlocal theory (Peridynamics) and Molecular Dynamic theories are discussed. Numerical examples related to large deformation and fracture problems are tested to verify the ability of the proposed method. The numerical results show that GPD inherits the advantages of the classical method and also has its unique properties in solid fracture mechanics and large deformation problems, making it a promising approach in computational mechanics.

Numerical Recovering of Space-Dependent Sources in Hyperbolic Transmission Problems

A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed.

Meshless Generalized Finite Difference Method Based on Nonlocal Differential Operators for Numerical Simulation of Elastostatics

This study proposes an innovative meshless approach that merges the peridynamic differential operator (PDDO) with the generalized finite difference method (GFDM). Based on the PDDO theory, this method introduces a new nonlocal differential operator that aims to reduce the pre-assumption required for the PDDO method and simplify the calculation process. By discretizing through the particle approximation method, this technique proficiently preserves the PDDO’s nonlocal features, enhancing the numerical simulation’s flexibility and usability. Through the numerical simulation of classical elastic static problems, this article focuses on the evaluation of the calculation accuracy, calculation efficiency, robustness, and convergence of the method. This method is significantly stronger than the finite element method in many performance indicators. In fact, this study demonstrates the practicability and superiority of the proposed method in the field of elastic statics and provides a new approach to more complex problems.

Phase-field modeling of anisotropic crack propagation based on higher-order nonlocal operator theory

This paper presents a novel higher-order nonlocal operator theory for the phase-field modeling of brittle fracture in anisotropic materials. Incorporating higher order nonlocal operators can enhance the accuracy of the phase-field model by effectively capturing long-range interactions that hold significance in numerous materials. The reproducing kernel particle method is employed to derive a nonlocal differential operator to enhance computational stability and accuracy. Moreover, the proposed method eliminates the need for direct computation of derivatives of the modified kernel function, which avoids the calculation of moment matrix derivatives and improves computational efficiency. The phase-field modeling of polycrystalline materials, considering the anisotropic fracture resistance of each grain, is implemented using this numerical framework. The present method is able to capture different scenarios intergranular and transgranular crack propagation patterns in polycrystalline materials. The proposed method involves a detailed representation of the complex process of crack initiation and propagation in 2D and 3D models of polycrystalline materials.

Coupled total- and semi-Lagrangian peridynamics for modelling fluid-driven fracturing in solids

This paper presents a novel computational approach for modelling fluid-driven fracturing in quasi-brittle solids using peridynamics. The approach leverages a rigorous coupling of the total- and semi-Lagrangian formulations of peridynamics. Specifically, the total-Lagrangian formulation, whch is based on classical peridynamics theory used for modelling fractures in solids, is combined with the semi-Lagrangian formulation to solve the Navier-Stokes equation using a non-local differential operator for weakly compressible fluid flow. The proposed approach offers a unified peridynamics-based framework that enables simulations of a wide range of fluid-driven fracturing problems in solids. The framework can model the solid as either an ordinary or non-ordinary material and quantify the fluid with different equations of state. To prevent unphysical inter-penetration, a fluid-solid interaction scheme that assumes fictitious fluid points in the solid domain is proposed to quantify the forces at the fluid-solid interface. The proposed computational approach is validated through simulations of several classic problems, including the dam break and the Kristianovich-Geertsma-de Klerk (KGD) problem. The predictive capability of the proposed approach is further demonstrated by numerical examples of fractures induced by fluid injection in sandstone, which reasonably capture the fracture patterns compared to experimental results. The presented approach offers an alternative to explicit modelling of fluid-driven fracturing processes and may find wide and useful applications to a variety of industrial and geophysical processes.

Fractional order mathematical model of Ebola virus under Atangana–Baleanu–Caputo operator

The aim of this paper is to analyze a fractional model of the Ebola virus. This study is important because it contributes to our understanding of the Ebola virus transmission dynamics using the notion of non-local differential operators. We aim to apply the recently implemented Atangana–Baleanu–Caputo (ABC) fractional derivative with the Mittag-Leffler kernel to study the Ebola virus model closely. The Picard–Lindelof approach is used to do a comprehensive study of the existence and uniqueness of the model’s solutions. The approximate solutions of the fractional order Ebola virus model were obtained using a numerical technique with the ABC operator, a combination of the fundamental theorem of fractional calculus and the two-step Lagrange polynomial interpolation. This innovative approach may offer new insights into the Ebola virus model that were not previously explored. Finally, the numerical simulations illustrate how the control parameters impact specific compartments within the model. The geometrical representation gives significant information about the model’s complexity and reliable information about the model. We simulate each model compartment at various fractional orders and compare them with integer-order simulations, highlighting the effectiveness of modern derivatives. The fractional analysis underscores the enhanced accuracy of non-integer order derivatives in capturing the Ebola virus model’s dynamics.

Theoretical and numerical analysis of a chaotic model with nonlocal and stochastic differential operators

A set of nonlinear ordinary differential equations has been considered in this paper. The work tries to establish some theoretical and analytical insights when the usual time-deferential operator is replaced with the Caputo fractional derivative. Using the Caratheodory principle and other additional conditions, we established that the system has a unique system of solutions. A variety of well-known approaches were used to investigate the system. The stochastic version of this system was solved using a numerical approach based on Lagrange interpolation, and numerical simulation results were produced.

Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators

Laplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion techniques are then used to convert the obtained solution from Laplace domain into time domain. Various famous methods for numerical inversion of Laplace transform are based on quadrature approximation of Bromwich integral. The key features are the contour deformation and the choice of the quadrature rule. In this work, the Gauss–Hermite quadrature method and the contour integration method based on the trapezoidal and midpoint rule are tested and evaluated according to the criteria of applicability to actual inversion problems, applicability to different types of fractional differential equations, numerical accuracy, computational efficiency, and ease of programming and implementation. The performance and efficiency of the methods are demonstrated with the help of figures and tables. It is observed that the proposed methods converge rapidly with optimal accuracy without any time instability.

Heat- and Wave-Type Equations with Nonlocal Operators, I. Compact Lie Groups

Abstract We prove existence and uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a nonlocal (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty $. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave-type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat-type equation as well.

A survey of KdV-CDG equations via nonsingular fractional operators

<abstract><p>In this article, the Korteweg-de Vries-Caudrey-Dodd-Gibbon (KdV-CDG) equation is explored via a fractional operator. A nonlocal differential operator with a nonsingular kernel is used to study the KdV-CDG equation. Some theoretical features concerned with the existence and uniqueness of the solution, convergence, and Picard-stability of the solution by using the concepts of fixed point theory are discussed. Analytical solutions of the KdV-CDG equation by using the Laplace transformation (LT) associated with the Adomian decomposition method (ADM) are retrieved. The solutions are presented using 3D and surface graphics.</p></abstract>

The Comparisons Between Peridynamic Differential Operators and Nonlocal Differential Operators

The Comparisons Between Peridynamic Differential Operators and Nonlocal Differential Operators

Nonlinear fractional stochastic heat equation driven by Gaussian noise rough in space

In this paper we consider a class of nonlinear fractional stochastic heat equation∂∂tu(t,x)=Dδαu(t,x)+σ(u(t,x))W˙(t,x),(t,x)∈[0,T]×R, in (1+1)-dimension with T>0, where Dδα is a nonlocal fractional differential operator with α∈(1,2) and |δ|≤2−α. The diffusion coefficient σ(⋅) is a measurable function. W˙ is a Gaussian noise which is white in time and behaves as a fractional Brownian motion with Hurst index H satisfying 3−α4<H<12 in the space variable. Under some mild assumptions, we prove the existence and uniqueness of the mild solution in some function spaces. Along the way, we study the moment bounds for the solution and show that the solution is weakly full intermittent.

Construction of Eigenfunctions to One Nonlocal Second-Order Differential Operator with Double Involution

In this paper, we study the eigenfunctions to one nonlocal second-order differential operator with double involution. We give an explicit form of the eigenfunctions to the boundary value problem in the unit ball with Dirichlet conditions on the boundary. For the problem under consideration, the completeness of the system of eigenfunctions is established.

A fractional nonlocal elastic model for lattice wave analysis

This paper investigates the link between lattice elasticity and nonlocal continuum mechanics from one-dimensional and multi-dimensional wave propagation problems. Eringen (1983) closed the bridge between one-dimensional linear lattice elasticity (called Lagrange lattice) and nonlocal elastic continuum. The Born-Kármán wave dispersive properties of the one-dimensional lattice are accurately fitted by Eringen's nonlocal differential elastic law using asymptotic or phenomenological arguments. A fractional version of Eringen's nonlocal model based on the introduction of a fractional Laplacian for the nonlocal differential operators can improve the accuracy in matching the wave dispersive curve of lattice dynamics. A multi-dimensional generalization of Lagrange lattice for cubic crystals is the lattice of Gazis et al. composed of central and angular interactions. We show in this paper that the differential Eringen's nonlocal elasticity differs from the lattice-based nonlocal elasticity continuum. In particular, Eringen's nonlocal elasticity preserves the isotropic property, whereas the cubic lattice is only isotropic in the low frequency regimes. The fractional generalization of Eringen's model accurately fitted the wave dispersive properties of the cubic crystal in the first Brillouin zone. The fractional generalization of Eringen's model highlights an isotropic response in the low frequency regimes and a cubic symmetry in the high frequency regime.

A NONLOCAL MODELING FOR SOLVING TIME FRACTIONAL DIFFUSION EQUATION ARISING IN FLUID MECHANICS

This study mainly investigates new techniques for obtaining numerical solutions of time-fractional diffusion equations. The fractional derivative term is represented in the Lagrange operational sense. First, we describe the temporal direction of the considered model using the Legendre orthogonal polynomials. Moreover, to archive a full discretization approach a type of nonlocal method has been applied that is known as the nonlocal peridynamic differential operator (PDDO). The PDDO is based on the concept of peridynamic (PD) interactions by proposing the PD functions orthogonal to each term in the Taylor Series Expansion (TSE) of a field variable. The PDDO for numerical integration uses the vicinity of each point (referred to as the horizon, which does not need background mesh). The PDDO is exclusively described in terms of integration (summation) throughout the interaction domain. As a result, it is unsusceptible to singularities caused by discontinuities. It does, however, need the creation of PD functions at each node. We numerically investigate the stability, the convergence of the scheme, which verifies the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method.

On the differential representation and color-kinematics duality of AdS boundary correlators

The AdS boundary correlators and their dual correlation functions of boundary operators have been the main dynamic observables of the holographic duality relating a bulk AdS theory and a boundary conformal field theory. We show that tree-level AdS boundary correlators for generic states can be expressed as nonlocal differential operators of a certain structure acting on contact Witten diagrams. We further write the boundary correlators in a form that is very similar to flat space amplitudes, with Mandelstam variables replaced by certain combinations of single-state conformal generators, prove that all tree-level AdS boundary correlators have a differential representation, and detail the conversion of such differential expressions to position space. We illustrate the construction through the computation of the boundary correlators of scalars coupled to gluons and gravitons; when converted to position space, they reproduce known results. Color-kinematics duality and BCJ relations can be defined in analogy with their flat space counterparts, and are respected by the scalar correlators with a gluon exchange. We also discuss potential approaches to the double copy and find that its direct generalization may require nontrivial extensions.

A thermodynamic two-temperature model with distinct fractional derivative operators for an infinite body with a cylindrical cavity and varying properties

The current work presents a novel model in the field of thermoelasticity to study an infinitely cylindrical cavity solid medium with variable thermal properties. The governing equations are based on a two-temperature heat conduction model with local and non-local fractional differential operators (Caputo-Fabrizio (CF) and Atangana-Baleanu (AB)). The surface surrounding the cylindrical cavity is due to decreasing heat flow and transport at a constant speed. The effects of different fractional differential operators have been shown and analyzed by making some comparisons and presenting them in tables. The numerical results and figures show the effects of the fractional arrangement parameter and the heat flux velocity significantly on the stress and deformation patterns, but they have little effect on the dynamic temperature and conductivity changes. The thermoelastic model with two temperatures and fractional operators is suitable for describing the phenomenon of heat transfer within elastic and viscous media.

A nonlocal method for image shadow removal

This paper proposes a new model for image shadow removal. The model reformulates a recent osmosis model with nonlocal differential operators. This allows to benefit from distant pixels similarities and thus improves restoration results. Some properties of this model are established and discussed, making it suitable for our application. Experimental results show that the nonlocal model obtained very good qualitative and quantitative results compared with state-of-the-art techniques.

ON ROOT FUNCTIONS OF NONLOCAL DIFFERENTIAL SECOND-ORDER OPERATOR WITH BOUNDARY CONDITIONS OF PERIODIC TYPE

In this paper we consider one class of spectral problems for a nonlocal ordinary differential operator (with involution in the main part) with nonlocal boundary conditions of periodic type. Such problems arise when solving by the method of separation of variables for a nonlocal heat equation. We investigate spectral properties of the problem for the nonlocal ordinary differential equation Ly (x) ≡ −y 00 (x) + εy00 (−x) = λy (x), −1 &lt; x &lt; 1. Here λ is a spectral parameter, |ε| &lt; 1. Such equations are called nonlocal because they have a term y 00 (−x) with involutional argument deviation. Boundary conditions are nonlocal y 0 (−1) + ay0 (1) = 0, y (−1) − y (1) = 0. Earlier this problem has been investigated for the special case a = −1. We consider the case a 6= −1. A criterion for simplicity of eigenvalues of the problem is proved: the eigenvalues will be simple if and only if the number r = sqrt{p (1 − ε) / (1 + ε)} is irrational. We show that if the number r is irrational, then all the eigenvalues of the problem are simple, and the system of eigenfunctions of the problem is complete and minimal but does not form an unconditional basis in L2(−1, 1). For the case of rational numbers r, it is proved that a (chosen in a special way) system of eigen- and associated functions forms an unconditional basis in L2(−1, 1).

Fractional dynamics and analysis for a lana fever infectious ailment with Caputo operator

This study provides a deep insight into the complex process of the lana fever infectious disease model by employing the advantages of a non-local fractional operator called Caputo derivative. The fractional-type lana fever model possessing memory effect and hereditary properties allows us to comprehend the dynamics of the disease transmission in detail. For this purpose, we introduce some crucial theoretical and numerical results for the suggested disease model with the aid of the Caputo fractional derivative. The existence and uniqueness of the solution of lana fever system are shown through the fixed point theorem. Moreover, we find the most sensitive parameters under the forward sensitivity index in relation to reproduction number R0. Also, some basic properties of the lana fever model such as virus-free equilibrium point, reproduction number, virus endemic equilibrium point, local and global stability are presented by using the Caputo derivative. On the other hand, further analysis of the fractional version of the lana fever model shows that the disease-free equilibrium is locally asymptotically stable when R0<1, and also the endemic equilibrium of the model under investigation is globally asymptotically stable when R0>1. Additionally, numerical results are presented to observe the advantages of the non-local fractional differential operator of Caputo.