He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations

Bijan Krishna Saha; A. M. Mohiuddin; Sushanta Parua

doi:doi:10.11648/j.ijtam.20170306.11

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He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations

Bijan Krishna Saha, A. M. Mohiuddin, Sushanta Parua

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 6)

Received: 29 April 2017 Accepted: 8 October 2017 Published: 10 November 2017

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Abstract

In this paper, linear and non-linear Fredholm Integro-Differential Equations with initial conditions are presented. Aiming to find out an analytic and approximate solutions to linear and non-linear Fredholm Integro-Differential Equations, this paper presents a comparative study of He’s Homotopy perturbation method with other traditional methods namely the Variational iteration method (VIM), the Adomian decomposition method (ADM), the Series solution method (SSM) and the Direct computation method (DCM). Comparison of the applied methods of analytic solutions reveals that He’s Homotopy perturbation method is tremendously powerful and effective mathematical tool.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 6)
DOI	10.11648/j.ijtam.20170306.11
Page(s)	174-181
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), 2017. Published by Science Publishing Group

Keywords

Homotopy Perturbation, Variational Iteration, Adomian Decomposition, Series Solution, Direct Computation Method

References

[1]	A. J. Jerri, Introduction to Integral Equations with Applications, Marcel Dekker, New York, 1971.
[2]	A. M. Golberg, Solution Methods for Integral Equations: Theory and applications, Plenum Press, New York, 1979.
[3]	A. M. Wazwaz, A First Course in Integral Equations, World Scientific, 1997.
[4]	F. G. Tricomi, Integral Equations, Dover, 1982.
[5]	G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, 1994.
[6]	G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Model 13 (1992) 17_43.
[7]	J. Biazar, H. Ghazvini, He's variational iteration method for solving linear and non-linear systems of ordinary differential equations, Appl. Math. Comput.191 (2007) 287_297.
[8]	J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg.178 (1999) 257_262.
[9]	J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Internat. J. Non-linear Mech. 35 (2000) 37_43.
[10]	J. H. He, Homotopy perturbation method: A new non-linear analytical technique, Appl. Math. Comput.135 (2003) 73_79.
[11]	J. H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput.156 (2004) 527_539.
[12]	J. H. He, Thehomotopy perturbation method for non-linear oscillators with discontinuities, Appl. Math. Comput.151 (2004) 287_292.
[13]	J. H. He, Application of homotopy perturbation method to non-linear wave equations, Chaos Solitons Fractals 26 (2005) 695_700.
[14]	J. H. He, Homotopy perturbation method for bifurcation of non-linear problems, Int. J. Non-linear Sci. Numer. Simul.6 (2005) 207_208.
[15]	J. H. He, Periodic solutions and bifurcations of delay-differential equations, Phys. Lett.347 (2005) 228_230.
[16]	J. H. He, Limit cycle and bifurcation of non-linear problems, Chaos Solitons Fractals 26 (2005) 827_833.
[17]	J. H. He, Determination of limit cycles for strongly non-linear oscillators, Phys. Rev. Lett. 90 (2003) Art. No. 174301.
[18]	J. H. He, Variational iteration method-a kind of non-linear analytical technique: Some examples, Int. J. Nonlinear Mech. 34 (4) (1999) 699_708.
[19]	J. H. He, Variational iteration method Some recent results and new interpretations, J. Comput. Appl. Math. 207 (2007) 3_17.
[20]	LanXu, Variational iteration method for solving integral equations, Comput. Math. Appl. 54 (2007) 1071_1078.
[21]	M. Tatari, M. Dehghan, On the convergence of He's variational iteration method, J. Comput. Appl. Math. 207 (1) (2007) 121_128.
[22]	R. P. Kanwal, Linear Integral Differential Equations Theory and Technique, Academic Press, New York, 1971.
[23]	R. K. Miller, Nonlinear Volterra Integral Equations, W. A Benjamin, Menlo Park, CA, 1967.
[24]	Sh. Q. Wang, J. H. He, Variational iteration method for solving integro-differential equations, Phys. Lett. A 367 (2007) 188_191.

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APA Style

Bijan Krishna Saha, A. M. Mohiuddin, Sushanta Parua. (2017). He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations. International Journal of Theoretical and Applied Mathematics, 3(6), 174-181. https://doi.org/10.11648/j.ijtam.20170306.11

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ACS Style

Bijan Krishna Saha; A. M. Mohiuddin; Sushanta Parua. He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations. Int. J. Theor. Appl. Math. 2017, 3(6), 174-181. doi: 10.11648/j.ijtam.20170306.11

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AMA Style

Bijan Krishna Saha, A. M. Mohiuddin, Sushanta Parua. He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations. Int J Theor Appl Math. 2017;3(6):174-181. doi: 10.11648/j.ijtam.20170306.11

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@article{10.11648/j.ijtam.20170306.11,
  author = {Bijan Krishna Saha and A. M. Mohiuddin and Sushanta Parua},
  title = {He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {174-181},
  doi = {10.11648/j.ijtam.20170306.11},
  url = {https://doi.org/10.11648/j.ijtam.20170306.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170306.11},
  abstract = {In this paper, linear and non-linear Fredholm Integro-Differential Equations with initial conditions are presented. Aiming to find out an analytic and approximate solutions to linear and non-linear Fredholm Integro-Differential Equations, this paper presents a comparative study of He’s Homotopy perturbation method with other traditional methods namely the Variational iteration method (VIM), the Adomian decomposition method (ADM), the Series solution method (SSM) and the Direct computation method (DCM). Comparison of the applied methods of analytic solutions reveals that He’s Homotopy perturbation method is tremendously powerful and effective mathematical tool.},
 year = {2017}
}

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TY  - JOUR
T1  - He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations
AU  - Bijan Krishna Saha
AU  - A. M. Mohiuddin
AU  - Sushanta Parua
Y1  - 2017/11/10
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DO  - 10.11648/j.ijtam.20170306.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 174
EP  - 181
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170306.11
AB  - In this paper, linear and non-linear Fredholm Integro-Differential Equations with initial conditions are presented. Aiming to find out an analytic and approximate solutions to linear and non-linear Fredholm Integro-Differential Equations, this paper presents a comparative study of He’s Homotopy perturbation method with other traditional methods namely the Variational iteration method (VIM), the Adomian decomposition method (ADM), the Series solution method (SSM) and the Direct computation method (DCM). Comparison of the applied methods of analytic solutions reveals that He’s Homotopy perturbation method is tremendously powerful and effective mathematical tool.
VL  - 3
IS  - 6
ER  -

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Author Information

Bijan Krishna Saha

Department of Mathematics, The University of Barisal, Barisal, Bangladesh
A. M. Mohiuddin

Department of Mathematics, The University of Barisal, Barisal, Bangladesh
Sushanta Parua

Department of Mathematics, The University of Barisal, Barisal, Bangladesh

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Plain Text BibTeX RIS

APA Style

Bijan Krishna Saha, A. M. Mohiuddin, Sushanta Parua. (2017). He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations. International Journal of Theoretical and Applied Mathematics, 3(6), 174-181. https://doi.org/10.11648/j.ijtam.20170306.11

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ACS Style

Bijan Krishna Saha; A. M. Mohiuddin; Sushanta Parua. He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations. Int. J. Theor. Appl. Math. 2017, 3(6), 174-181. doi: 10.11648/j.ijtam.20170306.11

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AMA Style

Bijan Krishna Saha, A. M. Mohiuddin, Sushanta Parua. He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations. Int J Theor Appl Math. 2017;3(6):174-181. doi: 10.11648/j.ijtam.20170306.11

Copy | Download

@article{10.11648/j.ijtam.20170306.11,
  author = {Bijan Krishna Saha and A. M. Mohiuddin and Sushanta Parua},
  title = {He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {174-181},
  doi = {10.11648/j.ijtam.20170306.11},
  url = {https://doi.org/10.11648/j.ijtam.20170306.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170306.11},
  abstract = {In this paper, linear and non-linear Fredholm Integro-Differential Equations with initial conditions are presented. Aiming to find out an analytic and approximate solutions to linear and non-linear Fredholm Integro-Differential Equations, this paper presents a comparative study of He’s Homotopy perturbation method with other traditional methods namely the Variational iteration method (VIM), the Adomian decomposition method (ADM), the Series solution method (SSM) and the Direct computation method (DCM). Comparison of the applied methods of analytic solutions reveals that He’s Homotopy perturbation method is tremendously powerful and effective mathematical tool.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - He’s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations
AU  - Bijan Krishna Saha
AU  - A. M. Mohiuddin
AU  - Sushanta Parua
Y1  - 2017/11/10
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170306.11
DO  - 10.11648/j.ijtam.20170306.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 174
EP  - 181
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170306.11
AB  - In this paper, linear and non-linear Fredholm Integro-Differential Equations with initial conditions are presented. Aiming to find out an analytic and approximate solutions to linear and non-linear Fredholm Integro-Differential Equations, this paper presents a comparative study of He’s Homotopy perturbation method with other traditional methods namely the Variational iteration method (VIM), the Adomian decomposition method (ADM), the Series solution method (SSM) and the Direct computation method (DCM). Comparison of the applied methods of analytic solutions reveals that He’s Homotopy perturbation method is tremendously powerful and effective mathematical tool.
VL  - 3
IS  - 6
ER  -

Copy | Download