Oscillatory properties of certain nonlinear fractional nabla difference equations

In this paper, we investigate the oscillation of a class of nonlinear fractional nabla difference equations. Some oscillation criteria are established.     

Asymptotic behavior of nabla half order $h$-difference equations

In this paper we study the half order nabla fractional difference equation $ _{\rho(a)}\nabla^{0.5}_{h}x(t)=cx(t), ~ t\in(h\N)_{a+h},$ where $_{\rho(a)}\nabla^{0.5}_hx(t)$ denotes the Riemann-Liouville nabla half order $h$-difference of $x(t)$. We will establish the asymptotic behavior of the solutions of this equation satisfying $x(a)=A>0$ for various values of the constant $c$.     

Ulam‐Hyers stability of Caputo fractional difference equations

We study the Ulam‐Hyers stability of linear and nonlinear nabla fractional Caputo difference equations on finite intervals. Our main tool used is a recently established generalized Gronwall inequality, which allows us to give some Ulam‐Hyers stability results of discrete fractional Caputo equations. We present two examples to illustrate our main results.     

Gronwall's inequality for a nabla fractional difference system with a retarded argument and an application

ABSTRACT

We study the nabla fractional difference system with retarded argument. There are two major ingredients. A Gronwall's inequality for the nabla case is given. This allows us to evaluate the solution of nabla fractional difference system. We shall illustrate the validity of our results by means of examples.     

Lyapunov inequalities for nabla Caputo boundary value problems

ABSTRACT

We will establish uniqueness of solutions to boundary value problems involving the nabla Caputo fractional difference under two-point boundary conditions and give an explicit expression for the Green's functions for these problems. Using the Green's functions for specific cases of these boundary value problems, we will then develop Lyapunov inequalities for certain nabla Caputo BVPs.     

Riemann—Liouville型分数阶微分方程的微分变换方法

本文在Riemann-Liouville分数阶导数的广义Taylor公式的基础上,建立了求解Riemann-Liouville型分数阶微分方程的微分变换方法.本文所建立的基于Riemann-Liouville分数阶导数微分变换方法给求解Riemann-Liouville分数阶导数的微分方程提供了一种新工具。     

针对一维空间分数阶扩散方程的一个新的最大模近似公式（英文）

Based on the maximum principle,the difference formula defined on a non-integral node is given to approximate the fractional Riemann-Liouville derivative and the finite difference scheme for solving one-dimensional space fractional diffusion equations(FDEs) with variable coefficients is presented.Furthermore,using the maximum principle the scheme is proved unconditionally stable and secondorder accuracy in spatial grid size.Several numerical examples are given to verify the efficiency of the scheme.     

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0,1) or (1,2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.     

Solving homogeneous fractional differential equations of Euler type

We study linear homogeneous differential equations with three left Riemann-Liouville fractional derivatives; these equations are analogs of Euler ordinary differential equations. By using the direct and inverse Mellin transforms and residue theory, we obtain a complete system of linearly independent solutions. As a corollary, related results are proved for Euler ordinary differential equations.     

Comparison of Green's functions for a family of boundary value problems for fractional difference equations

ABSTRACT

In this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.     

Boundary value problem for a multidimensional fractional partial differential equation

We construct a fundamental solution of a linear fractional partial differential equation. For an equation with Dzhrbashyan-Nersesyan fractional differentiation operators, we solve a boundary value problem and find a closed-form representation for its solution. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are special cases of the assertions proved here.     

IMPROVED FRACTIONAL SUB-EQUATION METHOD AND ITS APPLICATIONS TO FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

Based on an improved fractional sub-equation method involving Jumarie's modified Riemann-Liouville derivative, we construct analytical solutions of space-time fractional compound Kd V-Burgers equation and coupled Burgers' equations. These results not only reveal that the method is very effective and simple in studying solutions to the fractional partial differential equation, but also include some new exact solutions.     

应用(G'/G~2)-展开法求解非线性分数阶微分方程（英文）

In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie's modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.     

Existence and Uniqueness of Solutions for the Initial Value Problem of Fractional $q_{k}$-Difference Equations for Impulsive with Varying Orders

The paper studies the existence and uniqueness for impulsive fractional $q_{k}$-difference equations of initial value problems involving Riemann-Liouville fractional $q_{k}$-integral and $q_{k}$-derivative by defining a new $q$-shifting operator. In this paper, we obtain existence and uniqueness results for impulsive fractional $q_{k}$-difference equations of initial value problems by using the Schaefer''s fixed point theorem and Banach contraction mapping principle. In addition, the main result is illustrated with the aid of several examples.     

一类含高阶Riemann-Liouville型分数阶导数脉冲微分方程的通解

本文首先运用迭代法获得一类含多项Riemann-Liouville型分数阶导数的微分方程的连续通解,然后应用数学归纳法得到这类脉冲微分方程的分片连续通解. 所得结果归结于脉冲分数阶微分方程领域,对分数阶微分方程研究者有参考意义.     

带非线性源项的双侧空间分数阶扩散方程的隐式中点方法

本文用隐式中点方法离散一阶时间偏导数,并用拟紧差分算子逼近Riemann-Liouville空间分数阶偏导数,构造了求解带非线性源项的空间分数阶扩散方程的数值格式.给出了数值方法的稳定性和收敛性分析.数值试验表明数值方法是有效的.     

Global attractivity for nonlinear fractional differential equations

We present some results for the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results.     

Positive solutions for a singular fractional boundary value problem

We investigate the existence of positive solutions for a system of Riemann-Liouville fractional differential equations, supplemented with uncoupled nonlocal boundary conditions which contain various fractional derivatives and Riemann-Stieltjes integrals, and the nonlinearities of the system are nonnegative functions and they may be singular at the time variable. In the proof of our main theorems, we use the Guo-Krasnosel'skii fixed point theorem.     

Solvability of an implicit fractional integral equation via a measure of noncompactness argument

In this paper, we study the existence of solutions to an implicit functional equation involving a fractional integral with respect to a certain function, which generalizes the Riemann-Liouville fractional integral and the Hadamard fractional integral. We establish an existence result to such kind of equations using a generalized version of Darbo's theorem associated to a certain measure of noncompactness. Some examples are presented.     

Fractional differential equations with worsening right-hand sides

We suggest methods for studying fractional differential equations with Caputo and Riemann-Liouville derivatives. Existence and uniqueness theorems are proved for the Cauchy problem for differential equations with worsening operators in scales of Banach spaces.