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

By Varsha Daftardar-Gejji, Hossein Jafari. 2005

359 citations recorded by the source corpus.

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: [ D α 1 y 1 , … , D α n y n ] t = A ( y 1 , … , y n ) t , y i ( 0 ) = c i , i = 1 , … , n , where A = [ a i j ] is a real square matrix, the solution turns out to be y ¯ ( x ) = E ( α 1 , … , α n ) , 1 ( x α 1 A 1 , … , x α n A n ) y ¯ ( 0 ) , where E ( α 1 , … , α n ) , 1 denotes multivariate Mittag-Leffler function defined for matrix arguments and A i is the matrix having ith row as [ a i 1 … a i n ] , and all other entries are zero. Fractional oscillation and Bagley–Torvik equations are solved as illustrative examples.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/34c2c5cecf8eb08a9e73f416712f27cd04e57ea5

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Citations in the corpus

Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives (2007, methodology): “Daftardar-Gejji and Jafari [3,7] have employed Adomian decomposition method to solve the linear/nonlinear systems of fractional differential equations which gives numerical answers to any order of desired accuracy.”
Novel solution methods for initial boundary value problems of fractional order with conformable differentiation (2017, background): “On the other hand, several researchers [5-17] have applied the homotopy perturbation/analysis methods (HPM/HAM) and Adomian decomposition method (ADM) to solve different kinds of fractional ordinary differential equations (ODEs), fractional partial differential equations (ODEs), integral equations (”
A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials (2012, methodology): “[9] H. Jafari, A. Kadem, D. Baleanu, T. Yilmaz, Solutions of the fractional Davey-Stewartson equations with variational iteration method.”
An iterative method for solving nonlinear functional equations (2006, methodology): “For definitions of Riemann–Liouville fractional integral and Caputo fractional derivative, we refer the reader to [6,12].”
Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models (2020, background): “The Bagley–Torvik equation has the same structure as equation (5) and is widely studied in the literature (Arqub and Maayah, 2018; Daftardar-Gejji and Jafari, 2005; Diethelm and Ford, 2002; Huang et al., 2016; Wang and Wang, 2010).”
A collocation-shooting method for solving fractional boundary value problems (2010, methodology): “For example, a generalized differential transform [28], is introduced to treat linear partial differential equations of fractional order, the He’s variational iteration method [14], the Adomian decomposition method [3,10,35] and the references therein.”
New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations (2014, background): “, [1–3]) are described by differential equations of fractional order.”
Solving Fractional Delay Differential Equations: A New Approach (2015, methodology): “methods for example, Adomian decomposition, homotopy perturbation and New iterative methods, have widely been employed in the recent literature [1, 11, 18, 5, 6, 2].”
Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation (2013, methodology): “Some other analytical methods and their recent development for solving nonlinear fractional partial differential equation can found in the work done by [15–26] and the excellent book for analytical and numerical methods.”
Solving a system of nonlinear fractional differential equations using Adomian decomposition (2006, methodology): “The present paper is a sequel to this work [7] and here Adomian method has been applied to a more general case incorporating nonlinearities as well, namely D i yi(x) = Ni(x, y1, .”
Solving a fourth‐order fractional diffusion‐wave equation in a bounded domain by decomposition method (2008, methodology): “Ginoa et al. [8] have presented a fractional diffusion equation describing relaxation Correspondence to: Hossein Jafari, Department of Mathematics and Computer Science, University of Mazandaran, Babolsar, Iran (e-mail: jafari_h@math.com) © 2007 Wiley Periodicals, Inc. phenomena in complex viscoelast”
The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative (2014, methodology): “There are analytical methods and numerical methods for solving the differential equations, such as the finite element method [6], the harmonic wavelet method [7–9], the Adomian decomposition method [10–12], the homotopy analysismethod [13, 14], the homotopy decomposition method [15, 16], the heat ba”