Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations

Joshua Ikechukwu Nwamba

doi:doi:10.11648/j.acm.20130203.12

| Peer-Reviewed

Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations

Joshua Ikechukwu Nwamba

Published in Applied and Computational Mathematics (Volume 2, Issue 3)

Received: 18 May 2013 Published: 30 June 2013

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The multi-order fractional Burgers-Poisson (MFBP) equation was introduced, exact as well as approximate solutions to the introduced MFBP, fractional Burgers-Poisson (fBP) and Burgers-Poisson (BP) equations were obtained through the use of the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM) in this paper. The effectiveness and efficiency of the approximate techniques in handling strongly nonlinear multi-order fractional as well as fractional partial differential equations was established in this paper. It was also shown in this paper that the two approximate techniques employed gave similar results to the considered model equations.

Published in	Applied and Computational Mathematics (Volume 2, Issue 3)
DOI	10.11648/j.acm.20130203.12
Page(s)	78-85
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), 2013. Published by Science Publishing Group

Keywords

Homotopy Perturbation Method, Adomian Decomposition Method, Burgers-Poisson Equation, Lie Method, Dispersive Media, Multi-Order Fractional Differential Equations

References

[1]	K. Fellnerand and C. Schmeiser. Burgers-Poisson: a nonlinear dispersive wave equation: SIAM J. Appl. Math. 64, 1509-1525, 2004.
[2]	N. C. Turgay and E. Hizel. Group Invariant Solutions for Burgers-Poisson Equation: Int. Math. Forum, 55, 2701-2710, 2007.
[3]	E. Hizel and S. Kucukarslan. A numerical analysis of Burgers-Poisson equation using variational iteration method: 3rd WSEAS ICATM, Spain, December, 14-16, 2007.
[4]	R. Fetecau and D. Levy. Approximate model equation for water waves: Comm. Math. Sci. 3, 159-170, 2005.
[5]	M. Kato. Large time behavior of solutions to the Burgers-Poisson equations: Osaka J. Math. 4(4), 923-943, 2007.
[6]	K. Hamer. Nonlinear effects on the propagation of sound waves in a radiating gas: Quart. J. Mech. Appl. Math. 24, 155-168, 1971.
[7]	W. G. Vincent and C. H. Kruger. Introduction to physical gas dynamics, Wiley, New York, 1965.
[8]	C. Zeng , Q. Yang and B. Zhang. Homotopy perturbation method for fractional-order Burgers-Poisson equation. Available from arXiv:1003.1828v1 [nlin.PS], 2010.
[9]	J. H. He. Homotopy Perturbation Technique: Comput. Meth. Appl. Mech. Eng. 178, 257-262, 1999.
[10]	J. H. He. An Elementary introduction to the homotopy perturbation method. Computers and Mathematics with Applications, 57(3), 410-412, 2009.
[11]	J. H. He. The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl.Math.Comput.151, 287-292, 2004.
[12]	J. H. He. Homotopy Perturbation Method: A new nonlinear analytical technique, Appl. Math. Comput. 137, 73-79, 2003.
[13]	J. H. He. Recent development of the homotopy perturbation method, Topols. Meth. Nonlinear Anal. 31, 205-209, 2008.
[14]	G. Adomian. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston 1994.
[15]	G. Adomian. Solutions of nonlinear PDE. Applied Mathematical Letters 11(13), 121-123, 1998.
[16]	P. Pue-on and N. Viriyapong. Modified Adomian Decomposition Method for Solving Particular Third-Order Ordinary Differential Equations. Applied Mathematical Sciences 6(30), 1463-1469, 2012.
[17]	R. Montazeri. A Concrete Application of Adomian Decomposition Method. Int. J. Contemp. Math. Sciences 7(24), 1185-1192, 2012.
[18]	H. Fatoorehchi and H. Abolghasemi. Adomian Decomposition Method to Study Mass Transfer from a Horizontal Flat Plate subject to Laminar Fluid Flow. Advances in Natural and Applied Sciences 5(1), 26-33, 2011.
[19]	J. I. Nwamba. Delayed Mathieu equation with fractional order damping: An approximate analytical solution. Int. J. Mech. Appl. 3 (4), 70-76, 2013.doi:105923/j.mechanics 20130304.02.
[20]	Arikoglu and I. Ozkol. Solution of fractional integro-differential equations by using fractional differential transform method: Chaos, Solitons and Fractals, 40, 521-529, 2009.
[21]	Ghorbani. Towards a new analytical method for solving fractional differential equations: Comp. Math. Appl. Mech. Eng., 197, 4173-4177, 2008.
[22]	M. M. Ghasemi. Numerical solution of the nonlinear integro-differential equations: Wavelet-Galerkin method and Homotopy perturbation method, Appl. Math. Comput., 188, 450-455, 2007.
[23]	Taiwo and O. S. Adetunde. Approximation of Multi-order fractional differential equations by an iterative decomposition method: American Journal of Engineering Science and Technology Research, 1(2), 10-18, 2013.
[24]	K. Hanan. The Homotopy analysis method for solving Multi-fractional order integro-differential equations: J. Al-Nahrain Univ., 14(3), 139-143, 2011.
[25]	Podlubny. Fractional Differential Equations, Academic press, San Diego, California, 1999.
[26]	J. West, M. Bolognab and P. Grigolini. Physics of Fractal operators, Springer, New York, 2003.
[27]	H. M. Srivasta, R.K. Saxena. Operators of fractional integration and their applications, Appl. Math. Comput., 118, 1-52, 2001.
[28]	R. L. Bagley, P. J. Torvik. A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27, 201-210, 1983.
[29]	R. Hilfer (ED). Applications of Fractional Calculus in Physics, Applications of Fractional Calculus in Physics, 2000.
[30]	Ross. A brief history and exposition of the fundamental theory of fractional calculus, In: Fractional calculus and its applications. Springer lecture notes in mathematics, 57, 1-36, 1975.

Cite This Article

Plain Text BibTeX RIS

APA Style

Joshua Ikechukwu Nwamba. (2013). Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations. Applied and Computational Mathematics, 2(3), 78-85. https://doi.org/10.11648/j.acm.20130203.12

Copy | Download

ACS Style

Joshua Ikechukwu Nwamba. Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations. Appl. Comput. Math. 2013, 2(3), 78-85. doi: 10.11648/j.acm.20130203.12

Copy | Download

AMA Style

Joshua Ikechukwu Nwamba. Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations. Appl Comput Math. 2013;2(3):78-85. doi: 10.11648/j.acm.20130203.12

Copy | Download

@article{10.11648/j.acm.20130203.12,
  author = {Joshua Ikechukwu Nwamba},
  title = {Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations},
  journal = {Applied and Computational Mathematics},
  volume = {2},
  number = {3},
  pages = {78-85},
  doi = {10.11648/j.acm.20130203.12},
  url = {https://doi.org/10.11648/j.acm.20130203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130203.12},
  abstract = {The multi-order fractional Burgers-Poisson (MFBP) equation was introduced, exact as well as approximate solutions to the introduced MFBP, fractional Burgers-Poisson (fBP) and Burgers-Poisson (BP) equations were obtained through the use of the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM)  in this paper. The effectiveness and efficiency of the approximate techniques in handling strongly nonlinear multi-order fractional as well as fractional partial differential equations was established in this paper. It was also shown in this paper that the two approximate techniques employed gave similar results to the considered model equations.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations
AU  - Joshua Ikechukwu Nwamba
Y1  - 2013/06/30
PY  - 2013
N1  - https://doi.org/10.11648/j.acm.20130203.12
DO  - 10.11648/j.acm.20130203.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 78
EP  - 85
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20130203.12
AB  - The multi-order fractional Burgers-Poisson (MFBP) equation was introduced, exact as well as approximate solutions to the introduced MFBP, fractional Burgers-Poisson (fBP) and Burgers-Poisson (BP) equations were obtained through the use of the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM)  in this paper. The effectiveness and efficiency of the approximate techniques in handling strongly nonlinear multi-order fractional as well as fractional partial differential equations was established in this paper. It was also shown in this paper that the two approximate techniques employed gave similar results to the considered model equations.
VL  - 2
IS  - 3
ER  -

Copy | Download

Author Information

Joshua Ikechukwu Nwamba

Department of Mathematics, University of Nigeria, Nsukka, 410001, Nigeria

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Joshua Ikechukwu Nwamba. (2013). Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations. Applied and Computational Mathematics, 2(3), 78-85. https://doi.org/10.11648/j.acm.20130203.12

Copy | Download

ACS Style

Joshua Ikechukwu Nwamba. Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations. Appl. Comput. Math. 2013, 2(3), 78-85. doi: 10.11648/j.acm.20130203.12

Copy | Download

AMA Style

Joshua Ikechukwu Nwamba. Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations. Appl Comput Math. 2013;2(3):78-85. doi: 10.11648/j.acm.20130203.12

Copy | Download

@article{10.11648/j.acm.20130203.12,
  author = {Joshua Ikechukwu Nwamba},
  title = {Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations},
  journal = {Applied and Computational Mathematics},
  volume = {2},
  number = {3},
  pages = {78-85},
  doi = {10.11648/j.acm.20130203.12},
  url = {https://doi.org/10.11648/j.acm.20130203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130203.12},
  abstract = {The multi-order fractional Burgers-Poisson (MFBP) equation was introduced, exact as well as approximate solutions to the introduced MFBP, fractional Burgers-Poisson (fBP) and Burgers-Poisson (BP) equations were obtained through the use of the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM)  in this paper. The effectiveness and efficiency of the approximate techniques in handling strongly nonlinear multi-order fractional as well as fractional partial differential equations was established in this paper. It was also shown in this paper that the two approximate techniques employed gave similar results to the considered model equations.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - Exact and Explicit Approximate Solutions to the Multi-Order Fractional Burgers-Poisson and Fractional Burgers-Poisson Equations
AU  - Joshua Ikechukwu Nwamba
Y1  - 2013/06/30
PY  - 2013
N1  - https://doi.org/10.11648/j.acm.20130203.12
DO  - 10.11648/j.acm.20130203.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 78
EP  - 85
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20130203.12
AB  - The multi-order fractional Burgers-Poisson (MFBP) equation was introduced, exact as well as approximate solutions to the introduced MFBP, fractional Burgers-Poisson (fBP) and Burgers-Poisson (BP) equations were obtained through the use of the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM)  in this paper. The effectiveness and efficiency of the approximate techniques in handling strongly nonlinear multi-order fractional as well as fractional partial differential equations was established in this paper. It was also shown in this paper that the two approximate techniques employed gave similar results to the considered model equations.
VL  - 2
IS  - 3
ER  -

Copy | Download