Semilinear Fractional Differential Equations Research Articles

Convergence of Runge–Kutta-based convolution quadrature for semilinear fractional differential equations

For solving the semilinear fractional differential equations with the nonsmooth force term, we construct a class of Runge–Kutta-based convolution quadrature. Moreover, we analyse the convergence of the proposed scheme. In addition, we employ the fast Runge–Kutta approximation to reduce the calculation cost. Finally, we give some numerical experiments to verify the theoretical results.

Singular solutions of a fractional Dirichlet problem in a punctured domain

Let D be a bounded regular domain in Rn (n ? 3) containing 0, 0 < ? < 2, and ? < 1. We take up in this article the existence and asymptotic behavior of a positive continuous solution for the following semi-linear fractional differential equation (??|D)?2 u = a(x)u?(x) in D\{0}, with the boundary Dirichlet conditions lim |x|?0 |x|n??u(x) = 0 and lim x??D ?(x)2??u(x) = 0, where (??|D)?/2 is the fractional Laplace associated to the subordinate killed Brownian motion process in D and ?(x) = dist ( x, ?D) denotes the Euclidean distance between x and ?D. The function a is a positive continuous function in D\{0}, which may be singular at x = 0 and/or at the boundary ?D satisfying some appropriate assumption related to Karamata class. More precisely, we shall prove the existence and global asymptotic behavior of a positive continuous solution on ?D\{0}. We will use some potential theory arguments and Karamata regular variation theory tools.

High-order schemes based on extrapolation for semilinear fractional differential equation

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α∈(1,2).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2).$$\\end{document} The error has the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big ( d_{3} \ au ^{3- \\alpha } + d_{4} \ au ^{4-\\alpha } + d_{5} \ au ^{5-\\alpha } + \\cdots \\big ) + \\big ( d_{2}^{*} \ au ^{4} + d_{3}^{*} \ au ^{6} + d_{4}^{*} \ au ^{8} + \\cdots \\big ) $$\\end{document} at any fixed time tN=T,N∈Z+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_{N}= T, N \\in {\\mathbb {Z}}^{+}$$\\end{document}, where di,i=3,4,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{i}, i=3, 4,\\ldots $$\\end{document} and di∗,i=2,3,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{i}^{*}, i=2, 3,\\ldots $$\\end{document} denote some suitable constants and τ=T/N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au = T/N$$\\end{document} denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2)$$\\end{document} is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.

Existence and Uniqueness Results of Fractional Differential Inclusions and Equations in Sobolev Fractional Spaces

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some notions and properties on set-valued maps and give some examples to explain our main results. We explore and use in this paper the fundamental properties of set-valued maps, which are needed for the study of differential inclusions. It began only in the mid-1900s, when mathematicians realized that their uses go far beyond a mere generalization of single-valued maps.

On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative

This work aims to study the existing results of mild solutions for a semi-linear Atangana-Baleanu-Caputo fractional differential equation with order $ 0 < \theta < 1 $ in an arbitrary Banach space. We rely on some arguments to present the mild solution to our problem in terms of an $ \theta $-resolvent family. Then we study the existence of this mild solution by using Krasnoselskii's fixed point theorem. Finally, we give an example to prove our results.

Stochastic controllability of semilinear fractional control differential equations

The purpose of our research is to identify some sufficient conditions for the stochastic controllability of fractional-order semilinear stochastic systems. The required outcomes are achieved by dividing the system under consideration into two systems: a linear stochastic system and a semilinear deterministic system. The key outcomes are achieved by employing the Schauder fixed point technique, nonlinear monotonicity, and keeping just fractional-order α∈(1/2,1). One example is presented for demonstrating the outcomes.

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

In the present paper, we study the existence and uniqueness of the solution for multi–term fractional boundary value problem under Riemann–Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi–term problem by using some notions and properties on set‐valued maps together with classical fixed point theorems due to Leray–Schauder. The cases when the set‐valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results.

Existence and partial approximate controllability of nonlinear Riemann–Liouville fractional systems of higher order

This article studies the existence and partial approximate controllability of higher order nonlocal semilinear fractional differential equations with Riemann–Liouville derivatives avoiding Lipschitz assumptions of nonlinear operator and nonlocal functions. To derive the existence result, we make approximate systems corresponding to the original system. For this, we construct the mild solutions in terms of fractional resolvent. Then, we prove the partial approximate controllability of the nonlinear system by using the obtained existence result. Finally, we give an example to illustrate the established theory.

UNSTABLE MANIFOLDS FOR FRACTIONAL DIFFERENTIAL EQUATIONS

We prove the existence of unstable manifolds for an abstract semilinear fractional differential equation Dαu(t) = Au(t) + f(u(t)), u(0) = u 0 , on a Banach space. We then develop a general approach to establish a semidiscrete approximation of unstable manifolds. The main assumption of our results are naturally satisfied. In particular, this is true for operators with compact resolvents and can be verified for finite elements as well as finite differences methods.

STEPANOV $\rho$-ALMOST PERIODIC FUNCTIONS IN GENERAL METRIC

In this paper, we analyze various classes of multi-dimensional Stepanov $\rho$-almost periodic functions in general metric. The main structural properties for the introduced classes of Stepanov almost periodic type functions are established. We also provide an illustrative application to the abstract degenerate semilinear fractional differential equations.

Existence and stability for a semilinear fractional differential equation with two delays

Abstract In this paper, we are concerned with a class of nonlinear fractional differential equation with delays. By means of the contraction mapping principle, we prove the existence of a unique solution and investigate the continuous dependence of the solution upon the initial data and two types of Ulam stability: Ulam-Hyers and Ulam-Hyers-Rassias ones. Then, we give an example to illustrate the main results.

Study on a semilinear fractional stochastic system with multiple delays in control

<abstract><p>This paper studies a semilinear fractional stochastic differential equation with multiple constant point delays in control. We transform the controllability problem into a fixed point problem. We obtain sufficient condition for the controllability by using Schauder's fixed point theorem. In addition, we discuss the optimal controllability of the problem. Some examples are given to illustrate the main result.</p></abstract>

Existence and Uniqueness of the Mild Solution of an Abstract Semilinear Fractional Differential Equation with State Dependent Nonlocal Condition

The purpose of this paper is to investigate the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract fractional differential equation with state dependent nonlocal condition. Continuous dependence of solutions on initial conditions and local ????-approximate mild solution of the considered problem will be discussed.

(ω, c)-periodic and asymptotically (ω, c)-periodic mild solutions to fractional Cauchy problems

In this paper, we establish some new properties of -periodic and asymptotically -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) and (2) where stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space and the function is -periodic or asymptotically -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations.

Weak asymptotic stability for semilinear fractional differential equations with finite delays

Weak asymptotic stability for semilinear fractional differential equations with finite delays

A new numerical method for solving semilinear fractional differential equation

A new numerical method for solving semilinear fractional differential equation

Hölder regularity for abstract semi-linear fractional differential equations in Banach spaces

In the present work the optimal regularity, in the sense of Hölder continuity, of linear and semi-linear abstract fractional differential equations is investigated in the framework of complex Banach spaces. This framework has been considered by the authors as the most convenient to provide a posteriori error estimates for the time discretizations of such a kind of abstract differential equations. In the spirit of the classical a posteriori error estimates, under certain assumptions, the error is bounded in terms of computable quantities, in our case measured in the norm of Hölder continuous and weighted Hölder continuous functions.

The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses

Abstract In this paper, we investigate a class of semilinear fractional differential equations with non-instantaneous impulses and integral boundary value conditions. By the method of upper and lower solutions combined with Amann three-solution theorem, existence results of at least three solutions are obtained.

Approximation of mild solutions of a semilinear fractional differential equation with random noise

Approximation of mild solutions of a semilinear fractional differential equation with random noise

Semilinear fractional stochastic differential equations driven by a γ-Hölder continuous signal with γ > 2/3

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a [Formula: see text]-Hölder continuous function [Formula: see text] with [Formula: see text]. Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to [Formula: see text] is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374].