Comments on the concept of existence of solution for impulsive fractional differential equations

In some recent works dealing with the existence of solutions for impulsive fractional differential equations, it is pointed out that the concept of solutions for such equations in some preceding papers is incorrect. In support of this claim, the authors of these papers begin with a counterexample. The objective of this note to indicate the mistake in these counterexamples and show the plausibility of the previous results.     

Operational solution of fractional differential equations

The main goal of this paper is to solve fractional differential equations by means of an operational calculus. Our calculus is based on a modified shift operator which acts on an abstract space of formal Laurent series. We adopt Weyl’s definition of derivatives of fractional order.     

Existence of fractional differential equations

Consider the fractional differential equation

Dαx=f(t,x),     

Existence results for fractional order functional differential equations with infinite delay

The Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay.     

Application of generalized differential transform method to multi-order fractional differential equations

In a recent paper [Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order, Appl Math Comput. submitted for publication] the authors presented a new generalization of the differential transform method that would extended the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form y^(μ)(t)=f(t,y(t),y^(β₁)(t),y^(β₂)(t),…,y^(β_n)(t)) with μ>β_n>β_n-1>…>β₁>0, combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization.     

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

In this paper, we study the numerical solution to time‐fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time‐fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of a system of fractional differential equations

We prove existence and uniqueness theorems for the initial value problem for the system of fractional differential equations , where D^α denotes standard Riemann-Liouville fractional derivative, 0<α<1, and A is a square matrix. The unique solution to this initial value problem turns out to be , where E_α denotes the Mittag-Leffler function generalized for matrix arguments. Further we analyze the system , , 0<α<1, and investigate dependence of the solutions on the initial conditions.     

Oscillations of neutral differential equations with variable coefficients

We obtain sufficient conditions for the oscillation of all solutions of the neutral differential equation d\dt [y(t) + P(t)y(t-T)] + Q(t)y(t-σ) = 0 where our results extend and improve several known results in the literature. These improvements and extensions are obtained by establishing and using some lemmas which are interesting in their own rights and which may have further applications in analysis.     

New existence results of positive solution for a class of nonlinear fractional differential equations

In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.     

Variational solutions for partial differential equations driven by a fractional noise

In this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D⊂R^d and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an L²(D)-valued fractional Wiener process W^H with Hurst parameter , whose covariance operator satisfies appropriate integrability conditions, and where γ∈(0,1] denotes the Hölder exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W^H.     

A sufficient condition of viability for fractional differential equations with the Caputo derivative

In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case.     

Positive solutions for a class of fractional differential equations with integral boundary conditions

The existence of positive solutions for a class of fractional equations involving the Riemann–Liouville fractional derivative with integral boundary conditions is investigated. By means of the monotone iteration method and some inequalities associated with the Green function, we obtain the existence of a positive solution and establish the iterative sequence for approximating the solution.     

A note on terminal value problems for fractional differential equations on infinite interval

In this note we establish sufficient conditions for existence and uniqueness of solutions of terminal value problems for a class of fractional differential equations on infinite interval. Some illustrative examples are comprehended to demonstrate the proficiency and utility of our results.     

Existence of solutions for fractional differential equations with multi-point boundary conditions

We discuss the existence of solutions for a nonlinear multi-point boundary value problem of integro-differential equations of fractional order q ∈ (1, 2]. Our analysis relies on the contraction mapping principle and the Krasnoselskii’s fixed point theorem. Example is provided to illustrate the theory.     

Difference methods for solving boundary value problems for fractional differential equations

Difference schemes for second-order ordinary and partial differential equations with a fractional time derivative are considered. Stationary and nonstationary problems for the diffusion equation in one-and multidimensional domains are examined separately. The stability and convergence of the difference schemes for these equations are proved.     

On the concept and existence of solution for impulsive fractional differential equations

This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

Numerical solutions to initial and boundary value problems for linear fractional partial differential equations

In this article, Haar wavelets have been employed to obtain solutions of boundary value problems for linear fractional partial differential equations. The differential equations are reduced to Sylvester matrix equations. The algorithm is novel in the sense that it effectively incorporates the aperiodic boundary conditions. Several examples with numerical simulations are provided to illustrate the simplicity and effectiveness of the method.