Error estimate of the series solution to a class of nonlinear fractional differential equations

El‐Kalla, I. L.

doi:10.1016/j.cnsns.2010.05.030

Cited by 36 publications

(48 citation statements)

References 17 publications

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“…By rearranging the terms of Taylor expansion of the nonlinear term f (u) about the zero component u 0 , the author in [3] deduced the same mathematical formula (5). In Sect.…”

Section: Introductionmentioning

confidence: 97%

“…ADM possesses a great potential in solving different kinds of functional equations. Application of ADM to different types of integral equations was discussed by many authors for example [6][7][8][9][10]. In this paper we consider the two dimensional nonlinear Fredholm-Volterra integral equation u(x, t) = f (x, t) + λ 1 t 0 k 1 (t, τ )f 1 (u(x, τ )) dτ + λ 2 b a k 2 (x, ζ )f 2 (u(t, ζ )) dζ.…”

Section: Introductionmentioning

confidence: 99%

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Error estimates for series solutions to a class of nonlinear integral equations of mixed type

El‐Kalla

2011

J. Appl. Math. Comput.

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In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225-234, 2005; Appl. Math. E-Notes 7: [214][215][216][217][218][219][220][221] 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian's accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.

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“…By rearranging the terms of Taylor expansion of the nonlinear term f (u) about the zero component u 0 , the author in [3] deduced the same mathematical formula (5). In Sect.…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

El‐Kalla

2011

J. Appl. Math. Comput.

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“…The author in [11,12] deduced a new formula to the Adomian's polynomials which can be written in the form…”

Section: Introductionmentioning

confidence: 99%

“…Formula (4) is called an accelerated Adomian polynomials and it was used successfully in [13] for solving a class of nonlinear fractional differential equations and in [14] for solving a class of nonlinear partial differential equations. Formula (4) has the advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations

Alkalla¹,

Abd-Elmonem²,

Gomaa³

2013

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In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian's series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.

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“…Both mathematicians and physicists have engaged in discussing the numerical methods for solving fractional differential equations. The most commonly used ones are Adomian decomposition method (ADM) [3,4], Variational iteration method (VIM) [5], Generalized differential transform method (GDTM) [6][7][8], Finite Difference Method (FDM) [9] and Wavelet method [10,11] and so on. In this paper, our study focuses on a class of fractional partial differential equation as following:…”

Section: Introductionmentioning

confidence: 99%

Chebyshev polynomials for approximation of solution of fractional partial differential equations with variable coefficients

Yang¹,

Hongbo²

2015

Proceedings of the 3rd International Conference on Material, Mechanical and Manufacturing Engineering

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Abstract. In this paper, a numerical method for solving a class of fractional partial differential equations with variable coefficients based on Chebyshev polynomials is proposed. The fractional derivative is described in the Caputo sense. The properties of Chebyshev polynomials are used to reduce the initial equations to the products of several matrixes. A system of linear equations are obtained by dispersing the coefficients and the products of matrixes. Only a small number of Chebyshev polynomials are needed to acquire a satisfactory result. Results obtained using the scheme presented here show that the numerical method is very effective and convenient for solving fractional partial differential equations with variable coefficients.

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Error estimate of the series solution to a class of nonlinear fractional differential equations

Cited by 36 publications

References 17 publications

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations

Chebyshev polynomials for approximation of solution of fractional partial differential equations with variable coefficients

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