Nonlocal Differential Equation Research Articles

Solving a non-local linear differential equation model of the Newtonian-type

Solving a non-local linear differential equation model of the Newtonian-type

General Solution to a Nonlocal Linear Differential Equation of First-Order

General Solution to a Nonlocal Linear Differential Equation of First-Order

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

Solvability and Controllability of a Retarded-Type Nonlocal Non-Autonomous Fractional Differential Equation

Solvability and Controllability of a Retarded-Type Nonlocal Non-Autonomous Fractional Differential Equation

Study of Nonlocal Multiorder Implicit Differential Equation Involving Hilfer Fractional Derivative on Unbounded Domains

This paper aims to study the existence and uniqueness of the solution for nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains a , ∞ , a ≥ 0 , in an applicable Banach space by utilizing the Banach contraction principle. Furthermore, we discuss various types of stability such as Ulam–Hyers–Rassias (UHR), Ulam–Hyers (UH), and semi-Ulam–Hyers–Rassias (sUHR) for nonlocal boundary value problem. Absolutely, our results cover that several outcomes have existed in the literature.

Estimates on the eigenvalues of complex nonlocal Sturm-Liouville problems

The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems. The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

<abstract><p>This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.</p></abstract>

Effective approximation for a nonlocal stochastic Schrödinger equation with oscillating potential

We study the effective approximation for a nonlocal stochastic Schrödinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit behavior as the scaling parameter tends to 0. This is motivated by data assimilation with non-Gaussian uncertainties. The nonlocal operator in this stochastic partial differential equation is the generator of a non-Gaussian Lévy-type process (i.e., a class of anomalous diffusion processes), with non-integrable jump kernel. With the help of a two-scale convergence technique, we establish effective approximation for this nonlocal stochastic partial differential equation. More precisely, we show that a nonlocal stochastic Schrödinger equation has a nonlocal effective equation. We show that it approximates the original stochastic Schrödinger equation weakly in a Sobolev-type space and strongly in \(L^2\) space. In particular, this effective approximation holds when the nonlocal operator is the fractional Laplacian.

The Flow of Polynomial Roots Under Differentiation

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian $$(-\Delta )^{1/2}$$ .

New existence results on nonlocal neutral fractional differential equation in concepts of Caputo derivative with impulsive conditions

In this article, we examine a nonlinear impulsive neutral fractional differential equation with nonlocal condition in an arbitrary Hilbert space. We obtain an associated integral equation and then consider a sequence of approximate integral equations by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The existence and uniqueness of an approximate solution by using analytic semigroup theory and the Fixed-point method are demonstrated. Convergence of the solutions of the approximate integral equation is proved. The Faedo-Galerkin approximation of the solution is studied and demonstrated some convergence results. Lastly, the application is presented to illustrate the theory of the main results.

On Nonlocal Vertical and Horizontal Bending of a Micro-Beam

The vertical and horizontal bending of micro-beams subjected to axial compressive and transverse concentrated loadings is a common lateral deformation in micro-/nano-engineering that plays a significant role in the design and optimization of micro-/nano-devices. The present study aims to investigate it using the nonlocal theory. For this purpose, the simplified mathematical model is developed, and the nonlocal differential constitutive equation is applied. Since the mechanical properties of micro-beams are different from those of macro-beams, the non-classical nonlocal bending moment is introduced to improve the classical bending formulation in order to adapt to the vertical and horizontal bending of micro-beams. The effects of the external load, external size, structural stiffness, and internal characteristic scale on the vertical and horizontal bending deformation including the midpoint deflection and critical compression are presented. The present analytical model and results are validated by the finite element method. It is shown that the critical compression decreases with increasing the internal characteristic scale. Moreover, the midpoint deflection varies remarkably with respect to the axial and transverse loadings, structural stiffness, internal characteristic scale, and external size. An obvious nonlocal scale effect is found, in which the internal characteristic scale cannot be neglected compared with the external size. Besides, a threshold value of the structural stiffness is determined, and the connotation of the nonlocal interaction requires that the structural stiffness shall not be lower than that threshold. A mutual restriction between structural stiffness and external loadings is observed in the vertical and horizontal bending. In particular, it is further proved that the classical continuum mechanics can not be used in micro-/nano-scaled mechanics through a strange phenomenon that is contrary to mechanical common sense in the calculation example. The study is expected to be beneficial to the design and application of micro-beams subjected to the vertical and horizontal bending.

Global Regularity for a Nonlocal PDE Describing Evolution of Polynomial Roots Under Differentiation

In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Steinerberger [Proc. Amer. Math. Soc., 147 (2019), pp. 4733--4744] to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to describe collective behavior of agents (such as birds, fish, or robots) in mathematical biology. We consider a periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data.

Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology

A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme.

Nonlocal Fractional Differential Equation On The Half Line in Banach Space

Our aim in this paper is to study the existence of solution sets and its topological structure for non-localfractional differential equations on the half-line in a Banach space using Riemann-Liouville definition. Themain result is based on Meir-Keeler fixed point theorem for condensing operators combined with measure ofnon-compactness. An example is given to illustrate the feasibility of our main result.

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 < x < 1. Here λ is a spectral parameter, |ε| < 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).

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

Asymmetrical cell division with exponential growth

An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to "exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal. doi:10.1017/S1446181121000109

Differential equations with multiple sign changing convolution coefficients

For continuous functions [Formula: see text] and [Formula: see text], which are allowed to change sign, we consider the nonlocal differential equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] represents the finite convolution of the functions [Formula: see text] and [Formula: see text]. A model case is the equation [Formula: see text] The existence of at least one positive solution to these problems subjected to a variety of boundary conditions is studied. Due to the use of a nonstandard order cone we are able to achieve our results without having to assume that the coefficients [Formula: see text] and [Formula: see text] are strictly positive.

Battery discharging model on fractal time sets

AbstractThis article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.

Nonlocal differential equations with concave coefficients of convolution type

For a continuous function A:R→R the existence of at least one positive solution of the nonlocal differential equation −A∫01(g∘u)(s)dsu′′(t)=λf(t,u(t)),t∈(0,1)is considered in the specific case in which g is a concave, strictly increasing continuous function. As a model case we consider −A∫01|u(s)|qdsu′′(t)=λf(t,u(t)),t∈(0,1),where 0<q<1. The results rely on the use of a nonstandard order cone together with an application of topological fixed point theory. Due to this the coefficient function A is neither required to be strictly positive nor monotone nor satisfy any particular growth condition. As a consequence of the theory developed for the preceding problem we also provide results for the nonlocal equation −A((b∗(g∘u))(1))u′′(t)=λf(t,u(t)),t∈(0,1)where t↦(b∗(g∘u))(t) is a finite convolution.