Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method

Eman Ali Ahmed Ziada

doi:doi:10.11648/j.ijssam.20210604.11

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Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method

Eman Ali Ahmed Ziada

Published in International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 4)

Received: 19 September 2021 Accepted: 14 October 2021 Published: 28 October 2021

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Abstract

In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 4)
DOI	10.11648/j.ijssam.20210604.11
Page(s)	111-119
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Fractional Differential Equation, Adomian Method, Existence, Uniqueness, Error Analysis

References

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[4]	Sh. A. Abd El-Salam, and A. M. A. El-Sayed, (2007), On the stability of some fractional-order non-autonomous systems, Electronic Journal of Qualitative Theory of Differential Equations, 6, pp. 1-14.
[5]	A. M. A. El-Sayed, and Sh. A. Abd El-Salam, (2008), On the stability of a fractional-order differential equation with nonlocal initial condition, Electronic Journal of Qualitative Theory of Differential Equations, 29, pp. 1-8.
[6]	D. J. Evans, and K. R. Raslan, (2005), The Adomian decomposition method for solving delay differential equation, International Journal of Computer Mathematics, (UK), 82, pp. 49-54.
[7]	D. Zwillinger, (1997), Handbook of Differential Equations, Academic Press.
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[10]	A. M. A. El-Sayed, E. M. El-Mesiry and H. A. A. El-Saka, (2004), Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. and Appl. Math., 23, 1, pp. 33-54.
[11]	E. M. El-Mesiry, A. M. A. El-Sayed and H. A. A. El-Saka, (2005), Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. and Comput., 160, 3, 683-699.
[12]	E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka (2007), Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., pp. 542–553.
[13]	A. M. A. El-Sayed, Linear differential equations of fractional orders, J. Appl. Math. Comput., (1993), pp. 1-12.
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[18]	K. Abbaoui, and Y. Cherruault, (1994), Convergence of Adomian’s method applied to differential equations, Computers Math. Applic., 28, pp. 103-109.
[19]	Y. Cherruault, G. Adomian, K. Abbaoui, and R. Rach, (1995), Further remarks on convergence of decomposition method, International J. of Bio-Medical Computing., 38, pp. 89-93.
[20]	N. T. Shawaghfeh, (2002), Analytical approximate solution for nonlinear fractional differential equations, J. Appl. Math. Comput., 131, pp. 517-529.
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Eman Ali Ahmed Ziada. (2021). Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method. International Journal of Systems Science and Applied Mathematics, 6(4), 111-119. https://doi.org/10.11648/j.ijssam.20210604.11

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ACS Style

Eman Ali Ahmed Ziada. Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method. Int. J. Syst. Sci. Appl. Math. 2021, 6(4), 111-119. doi: 10.11648/j.ijssam.20210604.11

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AMA Style

Eman Ali Ahmed Ziada. Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method. Int J Syst Sci Appl Math. 2021;6(4):111-119. doi: 10.11648/j.ijssam.20210604.11

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@article{10.11648/j.ijssam.20210604.11,
  author = {Eman Ali Ahmed Ziada},
  title = {Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {4},
  pages = {111-119},
  doi = {10.11648/j.ijssam.20210604.11},
  url = {https://doi.org/10.11648/j.ijssam.20210604.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210604.11},
  abstract = {In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.},
 year = {2021}
}

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TY  - JOUR
T1  - Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method
AU  - Eman Ali Ahmed Ziada
Y1  - 2021/10/28
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210604.11
DO  - 10.11648/j.ijssam.20210604.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 111
EP  - 119
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210604.11
AB  - In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.
VL  - 6
IS  - 4
ER  -

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Author Information

Eman Ali Ahmed Ziada

Basic Science Department, Nile Higher Institute for Engineering & Technology, Mansoura, Egypt

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Eman Ali Ahmed Ziada. (2021). Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method. International Journal of Systems Science and Applied Mathematics, 6(4), 111-119. https://doi.org/10.11648/j.ijssam.20210604.11

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ACS Style

Eman Ali Ahmed Ziada. Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method. Int. J. Syst. Sci. Appl. Math. 2021, 6(4), 111-119. doi: 10.11648/j.ijssam.20210604.11

Copy | Download

AMA Style

Eman Ali Ahmed Ziada. Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method. Int J Syst Sci Appl Math. 2021;6(4):111-119. doi: 10.11648/j.ijssam.20210604.11

Copy | Download

@article{10.11648/j.ijssam.20210604.11,
  author = {Eman Ali Ahmed Ziada},
  title = {Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {4},
  pages = {111-119},
  doi = {10.11648/j.ijssam.20210604.11},
  url = {https://doi.org/10.11648/j.ijssam.20210604.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210604.11},
  abstract = {In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Solution of Nonlinear Fractional Differential Equations Using Adomain Decomposition Method
AU  - Eman Ali Ahmed Ziada
Y1  - 2021/10/28
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210604.11
DO  - 10.11648/j.ijssam.20210604.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 111
EP  - 119
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210604.11
AB  - In this paper, Adomian decomposition method (ADM) will apply to solve nonlinear fractional differential equations (FDEs) of Caputo sense. These type of equations is very important in engineering applications such as electrical networks, fluid flow, control theory and fractals theory. ADM give analytical solution in form of series solution so the convergence of the series solution and the error analysis will discuss. In addition, existence and uniqueness of the solution will prove. Some numerical examples will solve to test the validity of the method and the given theorems. A comparison of ADM solution with exact and numerical methods are given.
VL  - 6
IS  - 4
ER  -

Copy | Download