Solving a multi-order fractional differential equation using adomian decomposition

Daftardar-Gejji, Varsha; Jafari, Hossein

doi:10.1016/j.amc.2006.11.129

Cited by 179 publications

(102 citation statements)

References 13 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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“…Numerical solutions for various types of FDEs by applying several techniques were proposed, for instance, Adomian's decomposition method [2,3], the Taylor collocation method [4], the variational iteration method [5], the finite difference method [6,7] and the ultraspherical wavelets method [8,9]. In addition, orthogonal polynomials have been widely used for obtaining numerical solutions for different types of FDEs.…”

Section: Introductionmentioning

confidence: 99%

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Abd-Elhameed

Youssri

2016

Entropy

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Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.

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“…Many approximation and numerical techniques are utilized to determine the numerical solution of multi-order FDE [8,11]. The Boubaker polynomials were established for the first time by Boubaker [1,5,6] as a guide for solving a one-dimensional heat transfer equation and second order differential equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

2016

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Abstract:In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

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Solving a multi-order fractional differential equation using adomian decomposition

Cited by 179 publications

References 13 publications

Efficient solution of a vibration equation involving fractional derivatives

Efficient solution of a vibration equation involving fractional derivatives

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

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