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.     

Initial value problems for arbitrary order fractional differential equations with delay

By fixed point theory the nonlinear alternative of Leray–Schauder type, and the properties of absolutely continuous functions space, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results involving local and global solutions.     

A short note on the solvability of impulsive fractional differential equations with Caputo derivatives

In this paper, we prove the existence and non-existence of solutions to two impulsive fractional differential equations with strong or weak Caputo derivatives in Euclidean space, respectively.     

On a nonlinear distributed order fractional differential equation

We study the existence and the uniqueness of mild and classical solutions for a class of equations of the form . Such equations arise in distributed derivatives models of viscoelasticity and system identification theory. We also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations.     

Existence of solutions of initial value problems for nonlinear fractional differential equations

In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results.     

A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations

In this paper, based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ?$?$. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.     

Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations

By means of a monotone iterative technique, we establish the existence and uniqueness of the positive solutions for a class of higher conjugate-type fractional differential equation with one nonlocal term. In addition, the iterative sequences of solution and error estimation are also given. In particular, this model comes from economics, financial mathematics and other applied sciences, since the initial value of the iterative sequence can begin from an known function, this is simpler and helpful for computation.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

In this paper, a new numerical method for solving fractional differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet is first presented. An operational matrix of fractional order integration is derived and is utilized to reduce the initial and boundary value problems to system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

A computational matrix method for solving systems of high order fractional differential equations

In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution u(t)$\mathbf{u}(\mathbf{t})$ and the Chebyshev coefficient matrix A^(ν)${\mathbf{A}}^{(\nu )}$ of the fractional derivative u^(ν)${\mathbf{u}}^{(\nu )}$. The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method.     

Nonlinear boundary value problem of first order impulsive functional differential equations

This paper discusses nonlinear boundary value problem for first order impulsive functional differential equations. We establish several existence results by using the lower and upper solutions and monotone iterative techniques. Two examples are discussed to illustrate the efficiency of the obtained results.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

On the existence of solutions for strongly nonlinear differential equations

The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations.     

New uniqueness results for fractional differential equations

We develop the Krasnoselskii–Krein type of uniqueness theorem for an initial value problem of the Riemann–Liouville type fractional differential equation which involves a function of the form f?(t,?x(t),?D ^q?1 x(t)), for 1<q<2 and establish the convergence of successive approximations. We prove a few other uniqueness theorems.     

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.     

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.     

Existence of fractional differential equations

Consider the fractional differential equation

Dαx=f(t,x),     

Solitary pattern solutions for fractional Zakharov-Kuznetsov equations with fully nonlinear dispersion

The fractional Zakharov-Kuznetsov equations are increasingly used in modeling various kinds of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. This has led to a significant interest in the study of these equations. In this work, solitary pattern solutions of fractional Zakharov-Kuznetsov equations are investigated by means of the homotopy perturbation method with consideration of Jumarie’s derivatives. The effects of fractional derivatives for the systems under consideration are discussed. Numerical results and a comparison with exact solutions are presented.     

A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation

This paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations. An algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed. Several illustrative examples, with linear and nonlinear fractional differential equations, are given to demonstrate the effectiveness of the method.     

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ (1, 2] by applying some standard fixed point theorems. An illustrative example is also presented.