Nonlocal Fractional Equations Research Articles

Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces

The maximal Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ B_{p,q}^{s}$$\\end{document} and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.

Existence Results for Kirchhoff Nonlocal Fractional Equations

Fractional and nonlocal operators of elliptic type arise in a quite natural way in many different contexts. In this paper, we study the existence of solutions for a class of fractional equations, while the nonlinear part of the problem admits some perturbation property. We obtain some new criteria for existence of two and infinitely many solutions, using critical point theory. Some recent results are extended and improved. Several examples are presented to demonstrate the applications of our main results.

Existence of the Mild Solution to Impulsive Nonlocal Fractional Integro-Differential Equations

Existence of the Mild Solution to Impulsive Nonlocal Fractional Integro-Differential Equations

On Some Fractional Magnetic Problems

In this article, we study existence of weak solutions and the unique continuation property of the nonlocal fractional magnetic equation. First, we use a variational technique to prove existence of weak solutions for the fractional Schrödinger equation with magnetic field. Moreover, we show the doubling property main argument to unique continuation property via Carleman estimates.

Multiplicity of solutions for nonlocal fractional equations with nonsmooth potentials

This paper is concerned a specific category of nonlocal fractional Laplacian problems that involve nonsmooth potentials. By utilizing an abstract critical point theorem for nonsmooth functionals and combining it with the analytical framework on fractional Sobolev spaces developed by Servadei and Valdinoci, we are able to establish the existence of at least three weak solutions for nonlocal fractional problems. Moreover, this work also generalizes and improves upon certain results presented in the existing literature.

Multiplicity of solutions for nonlocal fractional equations with nonsmooth potentials

This paper is concerned a specific category of nonlocal fractional Laplacian problems that involve nonsmooth potentials. By utilizing an abstract critical point theorem for nonsmooth functionals and combining it with the analytical framework on fractional Sobolev spaces developed by Servadei and Valdinoci, we are able to establish the existence of at least three weak solutions for nonlocal fractional problems. Moreover, this work also generalizes and improves upon certain results presented in the existing literature.

Thermoelastic response of an elastic rod under the action of a moving heat source based on fractional order strain theory considering nonlocal effects

In this paper, based on fractional order strain theory and nonlocal heat conduction theory, the thermoelastic response of an elastic rod rigidly fixed at both ends and subjected to a moving heat source is investigated. The nonlocal fractional order strain control equation is established, and the analytical expressions of dimensionless temperature, displacement, and stress are solved using the Laplace integral transformation and numerical inverse transformation. The distribution law of physical quantities such as temperature, displacement, and stress when the fractional parameters, heat source moving speed, and thermal nonlocal parameters change is obtained. The results show that strain parameters and thermal nonlocal parameters have less influence on thermal–mechanical waves, while heat source velocity has a significant influence on thermal–mechanical waves.

Symmetrical Solutions for Non-Local Fractional Integro-Differential Equations via Caputo–Katugampola Derivatives

Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra–Fredholm integro-differential equations, involving the Caputo–Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area.

The Existence, Uniqueness, and Multiplicity of Solutions for Two Fractional Nonlocal Equations

This paper establishes the existence of unique and multiple solutions to two nonlocal equations with fractional operators. The main results are obtained using the variational method and algebraic analysis. The conclusion is that there exists a constant λ*>0 such that the equations have only three, two, and one solution, respectively, for λ∈(0,λ*), λ=λ*, and λ>λ*. The main conclusions fill the gap in the knowledge of this kind of fractional-order problem.

Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non-homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.

Singular boundary behaviour and large solutions for fractional elliptic equations.

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equationsposed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equationwith zero boundarydata. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equationsdriven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Greenfunction.

Existence and uniqueness of the mild solution of an abstract nonlocal semilinear fractional integrodifferential equation

The purpose of this paper is to study the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract nonlocal fractional integrodifferential equation, which has a distinctive nonlinear term. Two sufficient conditions on the existence of mild solutions will be displayed. Continuous dependence of solutions on initial conditions and local [Formula: see text]-approximate mild solutions will be discussed. An example will be given to elucidate the main results. The results obtained are based upon the method of semigroups, contraction mapping principle, and Krasnoselskii’s fixed point theorem.

Nonlocal fractional elliptic equations and applications

AbstractThe Lp‐coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in Lp space and also is a generator of an analytic semigroup. Moreover, by using the Lp‐separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.

A discussion on controllability of nonlocal fractional semilinear equations of order [formula omitted] with monotonic nonlinearity

In this article, we use the tool of monotone nonlinearity to present the approximate controllability discussion for fractional semilinear system with nonlocal conditions. Monotonicity is an important characteristic in many communications applications in which digital-to-analog converter circuits are used. Such applications can function in the presence of nonlinearity, but not in the presence of non-monotonicity. Therefore, it becomes quite interesting to study a problem assuming monotonicity of the nonlinear function. Also, nonlocal conditions are additional specifications for the physical measurements than classical ones and impart a finer effect on the solution. We formulate the control function for the problem and establish the controllability results. At last, we proposed two applications for the demonstration.

On Positive Solution of a New Class of Nonlocal Fractional Equation with Integral Boundary Conditions

We prove that a positiveasolution to a given boundaryaproblem exists and is unique. This new boundaryacondition relates theanon-local unknown value of unknownafunction at λ with its influenceadue to aasup-strip (µ,1), 0<λ<µ<1 .Our resultsaare aobtainted by using " Banach andaKrasnoselskii’satheorems"a linked to anywhere. Some classicalatheorems of fixedapoints assistance to achieve the greatestaresults.

Regularity properties of nonlocal fractional differential equations and applications

Abstract The regularity properties of nonlocal fractional elliptic and parabolic equations in vector-valued Besov spaces are studied. The uniform B p , q s B_{p,q}^{s} -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. In particular, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, the maximal regularity properties of the nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional parabolic equations are studied.

Well-posed results for nonlocal fractional parabolic equation involving Caputo-Fabrizio operator

In this paper, we study the parabolic problem associated with non-local conditions, with the Caputo-Fabrizio derivative. Equations on the sphere have many important applications in physics, phenomena, and oceanography. The main motivation for us to study non-local boundary value problems comes from two main reasons: the first reason is that current major interest in several application areas. The second reason is to study approximation for the terminal value problem. With some given data, we prove that the problem has only the solution for two cases. In case \(\epsilon = 0,\) we prove the problem has a local solution. In case \(\epsilon > 0,\) then the problem has a global solution. The main tools and techniques in our demonstration are of using Banach's fixed point theorem in conjunction with some Fourier series analysis involved some evaluation of spherical harmonic function. Several upper and lower upper limit techniques for the Mittag-Lefler functions are also applied.

Existence results of mild solutions for nonlocal fractional delay integro-differential evolution equations via Caputo conformable fractional derivative

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauchy problem with Caputo conformable derivative in Banach spaces. We establish a representation of a mild solution using a fractional Laplace transform. The existence of such solutions is proved under certain conditions, using the Mönch fixed point theorem and a general version of Gronwall's inequality under weaker conditions in the sense of Kuratowski measure of non compactness. Applications illustrating our main abstract results and showing the applicability of the presented theory are also given.&lt;/p&gt;&lt;/abstract&gt;

Existence of mild solutions for nonlocal ψ−Caputo-type fractional evolution equations with nondense domain

Abstract The main crux of this manuscript is to establish the existence and uniqueness of solutions for nonlocal fractional evolution equations involving ψ−Caputo fractional derivatives of an arbitrary order α ∈ (0, 1) with nondense domain. The mild solutions of our proposed model are constructed by employing generalized ψ−Laplace transform and some new density functions. The proofs are based on Krasnoselskii fixed point theorem and some basic techniques of ψ−fractional calculus. As application, a nontrivial example is given to illustrate our theoritical results.

Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

In this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.