Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving Riemann-Liouville sequential fractional derivative by using monotone iterative method.     

The Hadamard Inequality For Convex Function Via Fractional Integrals

In this paper, we establish several inequalities for some differantiable mappings that are connected with the Riemann-Liouville fractional integrals. The analysis used in the proofs is fairly elementary.     

Some results on linear nabla Riemann-Liouville fractional difference equations

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann-Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann-Liouville fractional difference equations.     

Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators

Let b be a BMO function, and the commutator of order k for the Weyl fractional integral. In this paper we prove weighted strong type (p, p) inequalities (p > 1) and weighted endpoint estimates (p = 1) for the operator and for the pairs of weights of the type (w, ), where w is any weight and is a suitable one-sided maximal operator. We also prove that, for weights, the operator is controlled in the L ^p (w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators . The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral This research has been partially supported by Spanish goverment Grant MTM2005-8350-C03-02. The first author was also supported by CONICET, ANPCyT and CAI+D-UNL. The second author was also supported by Junta de Andalucía Grant FQM 354.     

Nonnegative solution for singular nonlinear fractional differential equation with coefficient that changes sign

In this paper, we consider the existence of nonnegative solutions of initial value problem for singular nonlinear fractional differential equation

New Exact Solutions of Fractional Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations Using Fractional Sub-Equation Method

In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense.     

分数阶微分方程的比较定理

本文给出了非线性Riemann—Liouville分数阶微分方程和Caputo分数阶微分方程与相应的非线性Volterra积分方程的等价性,并在此基础上建立了分数阶微分方程的比较定理．     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

New Solutions of Three Nonlinear Space- and Time-Fractional Partial Differential Equations in Mathematical Physics

Motivated by the widely used ansätz method and starting from the modified Riemann-Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.     

Numerical Method for the Time Fractional Fokker-Planck Equation

In this paper, a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed. The analysis of local truncation error and the stability of this method are investigated. Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.