Adomian decomposition: a tool for solving a system of fractional differential equations

Daftardar-Gejji, Varsha; Jafari, Hossein

doi:10.1016/j.jmaa.2004.07.039

Cited by 279 publications

(139 citation statements)

References 8 publications

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“…However, as mentioned in the introduction, it has also been used by several authors for fractional differential equations [17][18][19][20][21][22][23]. As it will be also used in this paper, it is presented in the followings.…”

Section: Adomian Decomposition Methodsmentioning

confidence: 99%

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Efficient solution of a vibration equation involving fractional derivatives

Pálfalvi

2010

International Journal of Non-Linear Mechanics

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Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980's, and they grow more and more popular nowadays.However, their efficient numerical calculation is nontrivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included.

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Section: Adomian Decomposition Methodsmentioning

confidence: 99%

“…Essentially, it approximates the solution of a non-linear differential equation with a series of functions. The method 2 A c c e p t e d m a n u s c r i p t is getting into use for the solution of fractional differential equations [17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Efficient solution of a vibration equation involving fractional derivatives

Pálfalvi

2010

International Journal of Non-Linear Mechanics

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“…There are many studies on Adomian decomposition method (ADM) which can evaluate the solutions of FPDEs. The ADM has been applied to obtain approximate solutions of linear or nonlinear fractional differential equations, fractional ordinary differential equations (FODEs), FPDEs, integral and integro-differential equations [7,8,9,10,11,12,13,14,15]. Meanwhile, there are many applications of solution methods based on variational iteration method (VIM) to ordinary-partial differential equations and other research areas [16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Approximate-analytical solutions of cable equation using conformable fractional operator

Yaşkıran¹,

Yavuz²

2017

ntmsci

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Abstract:In the present work, we have introduced a new formulation for the approximate-analytical solution of the fractional onedimensional cable differential equation (FCE) by using the conformable fractional derivative. First of all, we have redefined Adomian decomposition method (CADM) and variational iteration method (CVIM) in the conformable sense. Then, we have solved by using the mentioned methods, which can analytically solve the fractional partial differential equations (FPDEs). In order to represent the efficiencies of these proposed methods, we have compared the numerical and exact solutions of the (FCE). Also, we have found out that the proposed models defined with the conformable derivative operator are very efficient and powerful techniques in finding approximate-analytical solutions for the cable equation of fractional order. In addition, the classical derivative and integral properties are recovered partially when the fractional term (alpha) is equal to one.

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“…A numerical solution of problem (1)- (2) is given in [11,12] and analytical solutions in [10,13]. Papers [2,4,7,8] …”

Section: Introductionmentioning

confidence: 99%

“…Papers [2,7] state numerical solutions of problem (3) for α = 3/2 while [8] for α ∈ (1 2). In [4], analytical solutions of (3) are discussed in [1]. In the above Bagley-Torvik equations with fractional derivative of order α, α ∈ (1 2), it is assumed that A = 0.…”

Section: Introductionmentioning

confidence: 99%

Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

Staněk

2013

Open Mathematics

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We investigate the fractional differential equation) subject to the boundary conditions (0) = 0, (T )+ (T ) = 0. Here α ∈ (1 2), µ ∈ (0 1), is a Carathéodory function and c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives. MSC:34A08, 24A33, 34B15

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Adomian decomposition: a tool for solving a system of fractional differential equations

Abstract: Adomian decomposition method has been employed to obtain solutions of a system of fractional differential equations. Convergence of the method has been discussed with some illustrative examples. In particular, for the initial value problem:

Cited by 279 publications

References 8 publications

Efficient solution of a vibration equation involving fractional derivatives

Efficient solution of a vibration equation involving fractional derivatives

Approximate-analytical solutions of cable equation using conformable fractional operator

Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

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