In this article, by using Schaefer fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integro-differential equations with nonlocal conditions involving the Caputo fractional derivative.

In this paper, a new numerical method for solving a linear system of fractional integro-differential equations is presented. The fractional derivative is considered in the Caputo sense. The proposed technique is based on the new operational matrices of triangular functions. The suggested method reduces this type of system to the solution of system of linear algebraic equations. To demonstrate t...

This paper address a new vision for the generalized Mittag-Leffler stability of the fractional differential equations. We mainly focus on a new method, consisting of decomposing a given fractional differential equation into a cascade of many sub-fractional differential equations. And we propose a procedure for analyzing the generalized Mittag-Leffler stability for the given fractional different...

In this paper, we propose a method to approximate the solution of a linear Fredholm integro-differential equation by using the Chebyshev wavelet of the first kind as basis. For this purpose, we introduce the first Chebyshev operational matrix of integration. Chebyshev wavelet approximating method is then utilized to reduce the integro-differential equation to a system of algebraic equations. Il...

In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are des...

In this paper, Adomian decomposition method is applied to solve system linear fractional integro-differential equations. The fractional derivative is considered in the Caputo sense. Special attentions are given to study the convergence of the proposed method. Finally, some numerical examples are provided to show that this method is computationally efficient. Refer ences A. Arikoglu, and I. Ozko...

The purpose of this paper is to investigate some qualitative properties solutions nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo derivatives. These include uniform stability, asymptotic Mittag–Leffer stability and boundedness. presented results are proved by defining an appropriate Lyapunov function applying the Lyapunov–Razumikhin method (LRM). Henc...

This paper investigates the existence of solutions of the nonlinear fractional differential equation { Du(t) + f(t, u(t),Du(t)) = 0, 0 < t < 1, 3 < α ≤ 4, u(0) = u′(0) = u′′(0) = 0, u(1) = u(ξ), 0 < ξ < 1, where D is the Caputo fractional derivative, β > 0, α− β ≥ 1. The peculiarity of this equation is that the nonlinear term depends on the fractional derivative of the unknown function, compare...

In this paper, by using the theories and methods of integral equation and computer algebra, a reliable algorithm for solving the Volterra integral equation is established, and a new Maple algorithm mainproc is established, too. Some examples are presented to illustrate the implementations of the algorithm. The results of the examples indicate that the algorithm of Taylor polynomial method is si...