Existence and uniqueness of solution for fractional differential equations with integral boundary conditions and the Adomian decomposition method

Abstract

We propose an Adomian decomposition method to solve a class of nonlinear differential equations of fractional‐order with modified Caputo derivatives and integral boundary conditions. Our approach uses the integral boundary conditions to derive an equivalent nonlinear Volterra integral equation before establishing existence and uniqueness of solution and a recursion scheme for the solution. The convergence of the method is proved and an error analysis given. Two numerical examples are solved by obtaining a rapidly converging sequence of analytical functions to the solution.

New Stability Results for Abstract Fractional Differential Equations with Delay and Non-Instantaneous Impulses

“…In recent years, there has been significant progress in both ordinary and partial fractional differential equations. For more details on the applications of fractional calculus, the reader is directed to the books of Abbas et al [1][2][3], Herrmann [4], Hilfer [5], Kilbas et al [6], Samko et al [7], and Zhou [8] and papers [9][10][11][12][13][14][15]. In [16,17], Benchohra et al demonstrated the existence, uniqueness, and stability results for various classes of problems with different conditions and some form of extension of the well-known Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives.…”

On the Solutions of a Class of Nonlinear Fractional Differential Equations with Boundary Conditions

“…In most cases, we do not know the exact solution of the problem. Some numerical and approximate methods for solving fractional differential equations have been proposed, including the residual power series method, the homotopy perturbation method, fractional Adams-Moulton methods, variational interaction methods, and the Adomian decomposition method [3], [4], [9], [7], [10], [5], [15]. The Adomian decomposition method (ADM) was introduced by Adomian in the 1980s [2], [3], [4].…”

Editorial of the special issue on modelling, analysis, and applications

“…The Adomian decomposition method is used to solve a class of nonlinear differential equations of fractional order, with modified Caputo derivatives and integral boundary conditions. The latter are applied in the derivation of an equivalent nonlinear Volterra integral equation [16]. The dynamics of invasive-invaded species problems is introduced in another paper, based on a fourth-order parabolic operator, together with coupled nonlinear reaction terms [17].…”