Adomaian decomposition method for solving impulsive fractional differential equations

In this paper we apply the adomian decomposition method for solving impulsive fractional differential equations with caputo derivative, this is a powerful method which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. At the end we illustrate the proposed method by an example.

1 Introduction [1] Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary noninteger order.The subject is as old as the differential calculus and goes back to times when Leibniz and Newton invented differential calculus.One owes to Leibniz in a letter to L'Hospital, dated September 30, 1695 [13], the exact birthday of the fractional calculus and the idea of the fractional derivative.The idea of fractional calculus and fractional order differential equations greatly attracted the attention of mathematicians, physicists and engineers.
Production and hosting by ISPACS GmbH.http://www.ispacs.com/journals/cacsa/2017/cacsa-00083/International Scientific Publications and Consulting Services Indeed, in recent years fractional differential concept have been applied in wide range such as rheology, viscoelasticity, electrochemistry, signal processing, dynamics of earthquakes, optics, geology, viscoelastic materials, converters, biosciences, proteins, medicine, economics, probability and statistics, astrophysics, chemical engineering, splines, tomography, fluid mechanics, electromagnetic waves, control of power electronic, chaotic dynamics, porous media, polymer physics, electrochemistry, statistical physics, thermodynamics, neural networks, etc. [7,9,10,11,15,16,17,20,21,22,23,25,26].One of the most intensely studied topics has become important in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics is the theory of impulsive integer order differential equations.The study of impulsive fractional differential equations was initiated in the 1960s by Milman and Myshkis [18,19].The study of the general theory may refer to books of Benchohra et al. [3], Lakshmikantham et al. [12], Samoilenko and Peresyuk [23], and the references there in.However, to our knowledge, in spite of a large number of recently formulated applied problems, the numerical method for solving them are far less advanced.In this paper we describe a simple but effective method for case of "impulsive" fractional differential equations.for this purpose we use the adomian decomposition method for the following impulsive fractional differential equations: , where denotes the Caputo fractional derivative of order  with the lower limit zero,  ∶ [0, ] ×  →  is jointly continuous.The structure of paper is organized as follows.In Section 2, we introduce notations, definitions, and some preliminary notions. in Section 3, we verify the solution of impulsive fractional differential equation by ADM and it's convergence is expressed.Finally, in section 4 an example is given to check the accuracy of the method.

Preliminaries and notations
In this chapater we introduce notations, definitions, and some preliminary notions.Consider a general impulsive fractional system where  Proof.[14] Definition 2.1.The Caputo derivative of order q with the lower limit 0 for a function f can be written as Where (. ) is the Euler gamma function.by definition1, Caputo's derivative of a constant is equal to zero [2].

Definition 2.2. Adomian decomposition method:
Let us consider the general nonlinear functional equation: 2) Where N and f are, respectively, operator and function given in convenient spaces.We are looking for a function u satisfying equation (2.2).We suppose that N is such that (2.2) admits a unique solution [6,8] in some well-adapted space.Adomian's technique allows us to find the solution of (2.2) as an infinite series  = ∑   ∞ =1 using the recurrent scheme written below:  0 =   1 =  0 () ⋮   =  −1 ( 0 , … ,  −1 ) ⋮ where the   's are a special kind of polynomial (called Adomian's polynomials) calculated owing to the basic formula:

Adomian decomposition method for impulsive fractional differential equations
Consider the impulsive fractional differential equation : By integrating on sense of caputo we have Relationship gives an approximate analytical solution which converges perfectly towards the exact solution in the limit where  → ∞ [4,6].

Numerical examples
In this section, we consider a first order impulsive differential equations: