A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations

Abdelilah Kamal H. Sedeeg

doi:doi:10.11648/j.ajmcm.20160101.12

| Peer-Reviewed

A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations

Abdelilah Kamal H. Sedeeg

Published in American Journal of Mathematical and Computer Modelling (Volume 1, Issue 1)

Received: 8 October 2016 Accepted: 20 October 2016 Published: 15 November 2016

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this article, we have proposed a reliable combination of Elzaki transform and homotopy perturbation method (ETHPM) to solve Nonlinear Fractional Heat -Like Equations. The nonlinear terms in the equations can be handled by using homotopy perturbation method (HPM). This methods is very powerful and efficient techniques for solving different kinds of linear and nonlinear fractional differential equations. The results reveal that the combination of ELzaki transform and homotopy perturbation method (ETHPM) ismore efficient and easier to handle when is compared with existing other methods in such PDEs.

Published in	American Journal of Mathematical and Computer Modelling (Volume 1, Issue 1)
DOI	10.11648/j.ajmcm.20160101.12
Page(s)	15-20
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), 2016. Published by Science Publishing Group

Keywords

Homotopy Decomposition Method, Nonlinear Fractional Heat-Like Equation, Elzaki Transform

References

[1]	K. B. Oldham and J. Spanier, “The Fractional Calculus”, Academic Press, New York, NY, USA, (1974).
[2]	I. Podlubny. “Fractional Differential Equations”, Academic Press, NewYork, NY, USA, (1999).
[3]	A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations”, Elsevier, Amsterdam, The Netherlands, (2006).
[4]	J. F. Cheng, Y. M. Chu, Solution to the linear fractional differential equation using Adomiandecomposition method, Mathematical Problems in Engineering, 2011, doi: 10.1155 /2011/587068.
[5]	J. H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinearproblems, International Journal of Non- Linear Mechanics, vol. 35, 2000, pp. 37-43.
[6]	J. H. He, New interpretation of homotopy perturbation method, International Journal ofModern Physics B, vol. 20, 2006b, pp. 2561-2668.
[7]	J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, (1999), pp. 257–262.
[8]	J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, vol. 167 (1-2), 1998, pp. 57–68.
[9]	EsmailHesameddini, Mohsen Riahi, HabibollaLatifizadeh, A coupling method of Homotopy technique and Laplace transform for nonlinear fractional differential equations, International Journal of Advances in Applied Sciences (IJAAS) Vol. 1, No. 4, December 2012, pp. 159~170.
[10]	AbdonAtangana and AdemKihcman, The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis, Volume 2013.
[11]	Rodrigue Batogna Gnitchogna, Abdon Atangana, Comparison of Homotopy Perturbation Sumudu Transform method and Homotopy Decomposition method for solving nonlinear Fractional Partial Differential Equations, Advances in Applied and Pure Mathematics.
[12]	Abdolamir Karbalaie, Mohammad Mehdi Montazer, Hamed Hamid Muhammed, New Approach to Find the Exact Solution of Fractional Partial Differential Equation, WSEAS TRANSACTIONS on MATHEMATICS, Issue 10, Volume 11, October 2012.
[13]	M. Khalid, Mariam Sultana, FaheemZaidi and UroosaArshad, Application of Elzaki Transform Method on Some Fractional Differential Equations, Mathematical Theory and Modeling, Vol. 5, No. 1, 2015.
[14]	AbdonAtangana and AdemKihcman, The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis, Volume 2013.
[15]	Eltayeb A. Yousif, Solution of Nonlinear Fractional Differential Equations Using the Homotopy Perturbation Sumudu Transform Method, Applied Mathematical Sciences, Vol. 8, 2014, no. 44, 2195 - 2210
[16]	G. Adomian, Solving frontier problems of physics: The decomposition method, Kluwer Academic Publishers, Boston and London, 1994.
[17]	J. S. Duan, R. Rach, D. Buleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in FractionalCalculus, vol. 3, no. 2, (2012). pp. 73–99.
[18]	Tarig M. Elzaki, (2011), The New Integral Transform “Elzaki Transform” Global Journal of Pure and Applied Mathematics, ISSN 0973-1768, Number 1, pp. 57-64.
[19]	Tarig M. Elzaki&Salih M. Elzaki, (2011), Application of New Transform “Elzaki Transform” to Partial Differential Equations, Global Journal of Pure and Applied Mathematics, ISSN 0973-1768, Number 1, pp. 65-70.
[20]	Tarig M. Elzaki&Salih M. Elzaki, (2011), On the Connections between Laplace and Elzaki transforms, Advances in Theoretical and Applied Mathematics, ISSN 0973-4554 Volume 6, Number 1, pp. 1-11.
[21]	Tarig M. Elzaki&Salih M. Elzaki, (2011), On the Elzaki Transform and Ordinary Differential Equation With Variable Coefficients, Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 6, Number 1, pp. 13-18.
[22]	Mohand M. AbdelrahimMahgoub, “On TheElzaki Transform of Heaviside Step Function with a Bulge Function,” IOSR Journal of Mathematics (IOSR-JM) Volume 11, Issue 2 Ver. III (Mar - Apr. 2015), PP 72-74.
[23]	Abdelbagy A. AlshikhandMohand M. AbdelrahimMahgoub, “A Comparative Study Between Laplace Transform and Two New Integrals “ELzaki” Transform and “Aboodh”Transform,” Pure and Applied Mathematics Journal 2016; 5 (5): 145-150.
[24]	Abdelbagy A. AlshikhandMohand M. AbdelrahimMahgoub, On The Relationship BetweenElzaki TransformAnd New IntegralTransform "ZZ Transform, International Journal of Development ResearchVol. 06, Issue, 08, pp. 9264-9270, August, 201.
[25]	T. M. Elzaki and E. M. A. Hilal, Solution of linear and nonlinear partial differential equations using mixture of Elzaki transform and the projected differential transform method, Math. Theo. & Model., 2 (2012), 50-59.
[26]	Sh. Chang, Il Chang, A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl. Math. & Compu. 195 (2008), 799-808.
[27]	G. C. Wu, “New trends in the variational iteration method,” Communications in Fractional Calculus, vol. 2, pp. 59–75, 2011.
[28]	G. C. Wu and D. Baleanu, “Variational iteration method for fractional calculus—a universal approach by Laplace transform,” Advances in Difference Equations, vol. 2013, article 18, 2013.

Cite This Article

Plain Text BibTeX RIS

APA Style

Abdelilah Kamal H. Sedeeg. (2016). A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations. American Journal of Mathematical and Computer Modelling, 1(1), 15-20. https://doi.org/10.11648/j.ajmcm.20160101.12

Copy | Download

ACS Style

Abdelilah Kamal H. Sedeeg. A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations. Am. J. Math. Comput. Model. 2016, 1(1), 15-20. doi: 10.11648/j.ajmcm.20160101.12

Copy | Download

AMA Style

Abdelilah Kamal H. Sedeeg. A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations. Am J Math Comput Model. 2016;1(1):15-20. doi: 10.11648/j.ajmcm.20160101.12

Copy | Download

@article{10.11648/j.ajmcm.20160101.12,
  author = {Abdelilah Kamal H. Sedeeg},
  title = {A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {1},
  number = {1},
  pages = {15-20},
  doi = {10.11648/j.ajmcm.20160101.12},
  url = {https://doi.org/10.11648/j.ajmcm.20160101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20160101.12},
  abstract = {In this article, we have proposed a reliable combination of Elzaki transform and homotopy perturbation method (ETHPM) to solve Nonlinear Fractional Heat -Like Equations. The nonlinear terms in the equations can be handled by using homotopy perturbation method (HPM). This methods is very powerful and efficient techniques for solving different kinds of linear and nonlinear fractional differential equations. The results reveal that the combination of ELzaki transform and homotopy perturbation method (ETHPM) ismore efficient and easier to handle when is compared with existing other methods in such PDEs.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations
AU  - Abdelilah Kamal H. Sedeeg
Y1  - 2016/11/15
PY  - 2016
N1  - https://doi.org/10.11648/j.ajmcm.20160101.12
DO  - 10.11648/j.ajmcm.20160101.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 15
EP  - 20
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20160101.12
AB  - In this article, we have proposed a reliable combination of Elzaki transform and homotopy perturbation method (ETHPM) to solve Nonlinear Fractional Heat -Like Equations. The nonlinear terms in the equations can be handled by using homotopy perturbation method (HPM). This methods is very powerful and efficient techniques for solving different kinds of linear and nonlinear fractional differential equations. The results reveal that the combination of ELzaki transform and homotopy perturbation method (ETHPM) ismore efficient and easier to handle when is compared with existing other methods in such PDEs.
VL  - 1
IS  - 1
ER  -

Copy | Download

Author Information

Abdelilah Kamal H. Sedeeg

Mathematics Department, Faculty of Education, Holy Quran and Islamic Sciences University, Khartoum, Sudan

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Abdelilah Kamal H. Sedeeg. (2016). A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations. American Journal of Mathematical and Computer Modelling, 1(1), 15-20. https://doi.org/10.11648/j.ajmcm.20160101.12

Copy | Download

ACS Style

Abdelilah Kamal H. Sedeeg. A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations. Am. J. Math. Comput. Model. 2016, 1(1), 15-20. doi: 10.11648/j.ajmcm.20160101.12

Copy | Download

AMA Style

Abdelilah Kamal H. Sedeeg. A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations. Am J Math Comput Model. 2016;1(1):15-20. doi: 10.11648/j.ajmcm.20160101.12

Copy | Download

@article{10.11648/j.ajmcm.20160101.12,
  author = {Abdelilah Kamal H. Sedeeg},
  title = {A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {1},
  number = {1},
  pages = {15-20},
  doi = {10.11648/j.ajmcm.20160101.12},
  url = {https://doi.org/10.11648/j.ajmcm.20160101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20160101.12},
  abstract = {In this article, we have proposed a reliable combination of Elzaki transform and homotopy perturbation method (ETHPM) to solve Nonlinear Fractional Heat -Like Equations. The nonlinear terms in the equations can be handled by using homotopy perturbation method (HPM). This methods is very powerful and efficient techniques for solving different kinds of linear and nonlinear fractional differential equations. The results reveal that the combination of ELzaki transform and homotopy perturbation method (ETHPM) ismore efficient and easier to handle when is compared with existing other methods in such PDEs.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - A Coupling Elzaki Transform and Homotopy Perturbation Method for Solving Nonlinear Fractional Heat-Like Equations
AU  - Abdelilah Kamal H. Sedeeg
Y1  - 2016/11/15
PY  - 2016
N1  - https://doi.org/10.11648/j.ajmcm.20160101.12
DO  - 10.11648/j.ajmcm.20160101.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 15
EP  - 20
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20160101.12
AB  - In this article, we have proposed a reliable combination of Elzaki transform and homotopy perturbation method (ETHPM) to solve Nonlinear Fractional Heat -Like Equations. The nonlinear terms in the equations can be handled by using homotopy perturbation method (HPM). This methods is very powerful and efficient techniques for solving different kinds of linear and nonlinear fractional differential equations. The results reveal that the combination of ELzaki transform and homotopy perturbation method (ETHPM) ismore efficient and easier to handle when is compared with existing other methods in such PDEs.
VL  - 1
IS  - 1
ER  -

Copy | Download