Nonlocal Operators Research Articles

SEMI-ANALYTICAL ANALYSIS OF A FRACTIONAL-ORDER PANDEMIC DYNAMICAL MODEL USING NON-LOCAL OPERATOR

A significant issue of a deadly virus, CoronaVirus (COV) disease, quickly spreadable among human beings is investigated in this paper. Therefore, fractional-ordered mathematical model representing transmission dynamics of SARS COV-2019 with four components, i.e., Susceptible [Formula: see text], Infected [Formula: see text], Recovered [Formula: see text] and Death [Formula: see text] is considered. The main target of this paper is to investigate the existence of solution, stability and their numerical simulations. Therefore, Attangana–Ballenu–Caputo (ABC) derivative is used as a tool to find the existence of solution, whereas Adomian decomposition in combination with the integral (Laplace) transformation (LADM) is utilized for their numerical approximations. Finally, the achieved analytical results are validated and visualized through numerical simulations using MATLAB for the better understanding of dynamics in the transmission of infection disease.

Nonlocal degenerate parabolic-hyperbolic equations on bounded domains

We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations \partial_{t}u+\operatorname{div}(f(u))=\mathcal{L}[b(u)] on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion operator \mathcal{L} can be any symmetric Lévy operator (e.g. fractional Laplacians) and b is nondecreasing and allowed to have degenerate regions ( b'=0 ). We propose an entropy solution formulation for the problem and show uniqueness of bounded entropy solutions under general assumptions. Existence of solutions is shown in a separate paper. The uniqueness proof is based on the Kružkov doubling of variables technique and incorporates several a priori results derived from our entropy formulation: an L^{\infty} -bound, an energy estimate, strong initial trace, weak boundary traces, and a nonlocal boundary condition. Our work can be seen as both extending nonlocal theories from the whole space to domains and local theories on domains to the nonlocal case. Unlike local theories, our formulation does not assume energy estimates. They are now a consequence of the formulation, but as opposed to previous nonlocal theories, they play an essential role in our arguments.

Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation

We study travelling wave solutions of a generalised Korteweg-de Vries-Burgers equation with a non-local diffusion term and a concave-convex flux. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional type derivative with order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of travelling waves that, formally, in the limit of vanishing diffusion and dispersion would give rise to non-classical shocks, that is, shocks that violate the Lax entropy condition. The proof is based on arguments that are typical in dynamical systems. The nature of the non-local operator makes this possible, since the resulting travelling wave equation can be seen as a delayed integro-differential equation. Thus, linearization around critical points, continuity with respect to parameters and a shooting argument, are the main steps that we have proved and adapted for solving this problem. For more information see https://ejde.math.txstate.edu/Volumes/2025/45/abstr.html

Non-local skew and non-local skew sticky Brownian motions

In this paper, we present a comprehensive study on the generalizations of skew Brownian motion and skew sticky Brownian motion by considering non-local operators at the origin for the heat equations on the real line. To begin, we introduce Marchaud-type operators and Caputo–Dzherbashian-type operators, providing an in-depth exposition of their fundamental properties. Subsequently, we describe the two stochastic processes and the associated equations. The non-local skew Brownian motion exhibits jumps, as a subordinator, at zero where the sign of the jump is determined by a skew coin. Conversely, the non-local skew sticky Brownian motion displays stickiness at zero, behaving as the inverse of a subordinator, resulting in non-Markovian dynamics.

Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach

Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach

ON THE SOLVABILITY OF DIRECT AND INVERSE PROBLEMS FOR A CLASS OF DEGENERATE PARABOLIC EQUATIONS WITH INVOLUTION

In this paper, for degenerate diffusion equations with involution, the solvability of the direct and inverse problems for determining the right-hand side is studied. The equation with a fractional derivative in the Caputo sense is considered. The elliptic part of the studied equation involves a nonlocal analogue of the Laplace operator with a coefficient depending on the time variable. By studying these problems with respect to the time variable, we obtain a one-dimensional degenerate equation with a fractional Caputo derivative. The solution of this equation is expressed by a special function of the Kilbas-Saigo type. Similarly, for the spatial variable, we obtain a spectral problem for the nonlocal Laplace operator with the Dirichlet boundary condition. We explicitly find the eigenfunctions and eigenvalues of this problem and show the completeness of the system of eigenfunctions in space. Using the classical Fourier method, solutions to the problems under consideration are sought in the form of expansions in a series of eigenfunctions. The absolute and uniform convergence of the series, the possibility of their differentiation term by term in all variables and the absolute and uniform convergence of the differentiated series are proved. The main statements concerning the problems considered are presented in the form of existence and uniqueness theorems.

On nonlocal problems with Neumann boundary conditions: scaling and convergence for nonlocal operators and solutions

Formulations of Neumann-type boundary conditions for boundary value problems in the nonlocal framework are beset with difficulties, some related to the choice of a proper scaling. Here we identify a space-dependent scaling for a nonlocal Neumann operator, for which we prove linear in δ (δ being the radius for the support for the kernel) convergence of the Neumann operator and O(δ2) convergence of solutions to their classical counterparts. The pointwise-like convergence of the nonlocal normal operator is cast as a new type of two-scale operator-point convergence, which we call condensated convergence. The results hold for general integrable kernels, a setting which is favored in numerical simulations. We support this analysis with numerical convergence studies using a piecewise linear discontinuous Galerkin discretization and show an O(δ2) rate of convergence of solutions, also exhibiting an O(h2) convergence, where h is the mesh size.

The Nehari manifold approach for Fractional a(.)-Laplacian problem: the Nehari manifold approach

Using the Nehari manifold, we establish the existence of two non-negative weak solutions for a fractional type problem driven by a non-local operator of the elliptic type in fractional Orlicz–Sobolev spaces. We show how the existence of solutions depends on the properties of the Nehari manifold. Moreover, under some suitable assumptions, continuous and compact embeddings results are established.

Modified Landweber iterative method for a backward problem in time of the diffusion equation with local and nonlocal operators

Modified Landweber iterative method for a backward problem in time of the diffusion equation with local and nonlocal operators

On complexity and duality

We explore the relationship between complexity and duality in quantum systems, focusing on how local and non-local operators evolve under time evolution. We find that non-local operators, which are dual to local operators under specific mappings, exhibit behaviour that mimics the growth of their local counterparts, particularly when considering state complexity. For the open transverse Ising model this leads to a neat organisation of the operator dynamics on either side of the duality, both consistent with growth expected in a quadratic fermion model like the Kitaev chain. When examining periodic chains, however, the mapping of boundary terms provides access to multiple branches of highly complex operators. These give rise to much larger saturation values of complexity for parity-mixing operators and are in contrast to what one would expect for a quadratic Hamiltonian. Our results shed light on the intricate relationship between non-locality, complexity growth, and duality in quantum systems.

Intrinsic Hölder spaces for fractional kinetic operators

We introduce anisotropic Hölder spaces that are useful for studying the regularity theory for non-local kinetic operators L, whose prototypical example is Lu(t,x,v)=∫RdCd,s|v-v′|d+2s(u(t,x,v′)-u(t,x,v))dv′+⟨v,∇x⟩+∂t,with (t,x,v)∈R×R2d. The Hölder spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure on R×R2d, with respect to which the operator L is invariant. We prove an intrinsic Taylor-like formula, whose remainder is bounded in terms of the anisotropic distance of the Galilean structure. Our achievements naturally extend analogous known results for purely differential operators on Lie groups.

Modeling for the menstrual cycle with non-local operators

Modeling for the menstrual cycle with non-local operators

Numerical approximation and simulation of a Volterra integro‐differential equation with a peridynamic differential operator

AbstractWe study the numerical approximation to a nonlocal Volterra integro‐differential equation, in which the integral term is the convolution product of a positive‐definite kernel and a nonlocal peridynamic differential operator (PDDO). Compared with the classical differential operators, the nonlocal PDDOs describe, for example, discontinuities and have demonstrated more widespread applications. The equation is discretized in space by the Galerkin finite element method, and we accordingly prove its error estimate. We then discretize the equation in time by the backward Euler method, and a positive quadrature rule is combined to approximate the convolution term. The convergence rate of the fully‐discrete finite element scheme is proved, and numerical experiments are carried out to substantiate the theoretical findings.

Gradient flow solutions for porous medium equations with nonlocal Lévy-type pressure

We study a porous medium-type equation whose pressure is given by a nonlocal Lévy operator associated to a symmetric jump Lévy kernel. The class of nonlocal operators under consideration appears as a generalization of the classical fractional Laplace operator. For the class of Lévy operators, we construct weak solutions using a variational minimizing movement scheme. The lack of interpolation techniques is ensued by technical challenges that render our setting more challenging than the one known for fractional operators.

Shape optimization problems involving nonlocal and nonlinear operators

Shape optimization problems involving nonlocal and nonlinear operators

Generalised symmetries and state-operator correspondence for nonlocal operators

We provide a one-to-one correspondence between line operators and states in four-dimensional CFTs with continuous 1-form symmetries. In analogy with 0-form symmetries in two dimensions, such CFTs have a free photon realisation and enjoy an infinite-dimensional current algebra that generalises the familiar Kac-Moody algebras. We construct the representation theory of this current algebra, which allows for a full description of the space of states on an arbitrary closed spatial slice. On \U0001d54a2 × \U0001d54a1, we rederive the spectrum by performing a path integral on \U0001d5393 × \U0001d54a1 with insertions of line operators. This leads to a direct and explicit correspondence between the line operators of the theory and the states on \U0001d54a2 × \U0001d54a1. Interestingly, we find that the vacuum state is not prepared by the empty path integral but by a squeezing operator. Additionally, we generalise some of our results in two directions. Firstly, we construct current algebras in (2p + 2)-dimensional CFTs, that are universal whenever the theory has a p-form symmetry, and secondly we provide a non-invertible generalisation of those higher-dimensional current algebras.

Asymptotic behavior of the basic reproduction number for periodic nonlocal dispersal operators and applications.

This paper is concerned with asymptotic behavior of the basic reproduction number defined by next generation nonlocal (convolution) dispersal operators in a time-periodic environment and applications. First we investigate the influence of the frequency and dispersal rate on the basic reproduction number, and we obtain that the basic reproduction number is monotone on the frequency. In the nonautonomous situation, the basic reproduction number is not a monotone function of dispersal rate in general. We derive the monotonicity for large frequency or dispersal rate. Then we apply the obtained results to a time-periodic SIS epidemic model and establish the existence and asymptotic profiles of the endemic periodic solution. Since solution maps of nonlocal system lack compactness, the standard uniform persistence theory and topological degree theory are unapplicable to obtain the existence of the endemic periodic solution. To overcome this difficulty, we apply the asymptotic fixed point theorem with the help of the Kuratowski measure of noncompactness.

Symmetry and monotonicity of solutions for a system

Abstract In our thesis, we employ a forth and means of traveling flats for an equation set including completely non-linear non-local operators in B1(0). Under the conditions ζ η ∈ C l o c 1 , 1 ∩ L α and υ η ∈ C l o c 1 , 1 ∩ L β , the symmetric character of radial direction and monotone character of plus solutions for an equation set are proved. In order to get this result, we use the extreme value theory of narrow area for the equation set and pivotal ingredients for carrying on the way of traveling flats.

More on unconstrained descriptions of higher spin massless particles

Here we suggest a new local action describing arbitrary integer spin-s massless particles in terms of only two symmetric fields φ and α of rank-s and (s−3) respectively. It is an unconstrained version of the Fronsdal theory where the double traceless constraint on the physical field is evaded via a rank-(s−4) Weyl like symmetry. The constrained higher spin diffeomorphism is enlarged to full diffeomorphism via the Stueckelberg field α through an appropriate field redefinition. After a partial gauge fixing where the Weyl symmetry is broken while preserving diffeomorphisms, the field equations reproduce, for arbitrary integer spin-s, diffeomorphism invariant equations of motion previously obtained via a truncation of the spectrum of the open bosonic string field theory in the tensionless limit. In the s=4 case we show that the functional integration over α leads to a unique nonlocal Weyl and diffeomorphism invariant action given only in terms of the physical field φ whose spectrum is confirmed via an analysis of the analytic structure of the spin-4 propagator for which we introduce a complete basis of projection and transition nonlocal differential operators. We also show that the elimination of α after the Weyl gauge fixing leads to a nonlocal diffeomorphism invariant action previously obtained in the literature. Published by the American Physical Society 2025

Calderón–Zygmund Estimates for the Fractional p-Laplacian

We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional p-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.