Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM

Kirtiwant Parshuram Ghadle; Khan Firdous; Khan Arshiya Anjum

doi:doi:10.11648/j.ajmcm.20170201.13

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Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM

Kirtiwant Parshuram Ghadle, Khan Firdous, Khan Arshiya Anjum

Published in American Journal of Mathematical and Computer Modelling (Volume 2, Issue 1)

Received: 30 October 2016 Accepted: 25 November 2016 Published: 13 January 2017

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Abstract

By handling the one dimensional partial differential equation with three methods i.e. Adomain decomposition method(ADM), Variation iteration method(VIM) and the New iterative method(NIM) and applied logarithmic and exponential functions as initial condition. A general framework of these methods is presented for analytical treatment of fractional partial differential equation arises in fluid mechanics. The fractional derivatives are described in the Caputo sense. The equation used in this paper is fractional wave equation, fractional burgers equation and fractional Klein-Gordon equation. After comparison of the results, the series of solution are found which is very helpful. The basic idea described in this paper is accepted to be further in use to solve other similar linear problems in fractional calculus.

Published in	American Journal of Mathematical and Computer Modelling (Volume 2, Issue 1)
DOI	10.11648/j.ajmcm.20170201.13
Page(s)	13-23
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

Adomain Decomposition Method (ADM), Variation Iteration Method (VIM), New Iterative Method (NIM), Fractional Wave Equation, Fractional Burgers Equation, Fractional Klein-Gordon Equation

References

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[2]	I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
[3]	V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations”, Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753-763, 2006.
[4]	A. A. Hemeda, “New iterative method: application to the nthorder integro-differential equations”, Information B, vol. 16, no. 6, pp. 3841-3852, 2013.
[5]	S. Bhalekar and V. Daftardar-Gejji, “New iterative method: application to partial differential equations”, Applied Mathematics and Computation, vol. 203, no. 2, pp. 778-783, 2008.
[6]	G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method”, Kluwer, 1994.
[7]	J. H. He, “Homotopy perturbation technique”, Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999.
[8]	A. A. Hemeda, “Variational iteration method for solving nonlinear coupled equations in 2-dimensional space in fluid mechanics”, International Journal of Contemporary Mathematical Sciences, vol. 7, no. 37, pp. 1839-1852, 2012.
[9]	J. Henderson, R. Luca, “Positive solution for a system of fractional differential equations with coupled integral boundary conditions”, Applied Mathematics and Computation, vol. 249, pp. 182-197, 2014.
[10]	A. Bekri, E. Aksoy and A. C. Cevikel, “Exact solutions of nonlinear time fractional partial differential equations by sub-equation method”, Math. Meth. Appl. Sci. vol. 38, pp. 2779-2784, 2015.
[11]	X-J. Yang, H. M. Srivastava and C. Cattani, “Local fractional homotopy perturbation method for solving fractional partial differential equations arising in mathematical physics”, Romanian Reports in Physics, vol. 67, no. 3, pp. 752-761, 2015.
[12]	A. H. Bhrawy, E. H. Doha and S. S. Ezz-Eldien, “A Spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations”, Journal of Computational Physics, vol. 293, pp. 142-156, 2015.
[13]	M. Caputo, “Linear models of dissipation whose Q is almost frequency independent part II”, Geophysical Journal International, vol. 13, no. 5, pp. 529-539, 1967.
[14]	I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation”, Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 367-386, 2002.
[15]	Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics”, Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2199-2208, 2009.
[16]	A. A. Hemeda, “Solution of fractional partial differential equations in fluid mechanics by extension of some iterative method”, Abstract and Applied Analysis, vol. 2013, pp. 9.

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APA Style

Kirtiwant Parshuram Ghadle, Khan Firdous, Khan Arshiya Anjum. (2017). Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM. American Journal of Mathematical and Computer Modelling, 2(1), 13-23. https://doi.org/10.11648/j.ajmcm.20170201.13

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ACS Style

Kirtiwant Parshuram Ghadle; Khan Firdous; Khan Arshiya Anjum. Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM. Am. J. Math. Comput. Model. 2017, 2(1), 13-23. doi: 10.11648/j.ajmcm.20170201.13

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AMA Style

Kirtiwant Parshuram Ghadle, Khan Firdous, Khan Arshiya Anjum. Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM. Am J Math Comput Model. 2017;2(1):13-23. doi: 10.11648/j.ajmcm.20170201.13

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@article{10.11648/j.ajmcm.20170201.13,
  author = {Kirtiwant Parshuram Ghadle and Khan Firdous and Khan Arshiya Anjum},
  title = {Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {1},
  pages = {13-23},
  doi = {10.11648/j.ajmcm.20170201.13},
  url = {https://doi.org/10.11648/j.ajmcm.20170201.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170201.13},
  abstract = {By handling the one dimensional partial differential equation with three methods i.e. Adomain decomposition method(ADM), Variation iteration method(VIM) and the New iterative method(NIM) and applied logarithmic and exponential functions as initial condition. A general framework of these methods is presented for analytical treatment of fractional partial differential equation arises in fluid mechanics. The fractional derivatives are described in the Caputo sense. The equation used in this paper is fractional wave equation, fractional burgers equation and fractional Klein-Gordon equation. After comparison of the results, the series of solution are found which is very helpful. The basic idea described in this paper is accepted to be further in use to solve other similar linear problems in fractional calculus.},
 year = {2017}
}

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TY  - JOUR
T1  - Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM
AU  - Kirtiwant Parshuram Ghadle
AU  - Khan Firdous
AU  - Khan Arshiya Anjum
Y1  - 2017/01/13
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170201.13
DO  - 10.11648/j.ajmcm.20170201.13
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  - 13
EP  - 23
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170201.13
AB  - By handling the one dimensional partial differential equation with three methods i.e. Adomain decomposition method(ADM), Variation iteration method(VIM) and the New iterative method(NIM) and applied logarithmic and exponential functions as initial condition. A general framework of these methods is presented for analytical treatment of fractional partial differential equation arises in fluid mechanics. The fractional derivatives are described in the Caputo sense. The equation used in this paper is fractional wave equation, fractional burgers equation and fractional Klein-Gordon equation. After comparison of the results, the series of solution are found which is very helpful. The basic idea described in this paper is accepted to be further in use to solve other similar linear problems in fractional calculus.
VL  - 2
IS  - 1
ER  -

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Author Information

Kirtiwant Parshuram Ghadle

Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India
Khan Firdous

Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India
Khan Arshiya Anjum

Department of Computer Science and Information Technology, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India

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Plain Text BibTeX RIS

APA Style

Kirtiwant Parshuram Ghadle, Khan Firdous, Khan Arshiya Anjum. (2017). Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM. American Journal of Mathematical and Computer Modelling, 2(1), 13-23. https://doi.org/10.11648/j.ajmcm.20170201.13

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ACS Style

Kirtiwant Parshuram Ghadle; Khan Firdous; Khan Arshiya Anjum. Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM. Am. J. Math. Comput. Model. 2017, 2(1), 13-23. doi: 10.11648/j.ajmcm.20170201.13

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AMA Style

Kirtiwant Parshuram Ghadle, Khan Firdous, Khan Arshiya Anjum. Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM. Am J Math Comput Model. 2017;2(1):13-23. doi: 10.11648/j.ajmcm.20170201.13

Copy | Download

@article{10.11648/j.ajmcm.20170201.13,
  author = {Kirtiwant Parshuram Ghadle and Khan Firdous and Khan Arshiya Anjum},
  title = {Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {1},
  pages = {13-23},
  doi = {10.11648/j.ajmcm.20170201.13},
  url = {https://doi.org/10.11648/j.ajmcm.20170201.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170201.13},
  abstract = {By handling the one dimensional partial differential equation with three methods i.e. Adomain decomposition method(ADM), Variation iteration method(VIM) and the New iterative method(NIM) and applied logarithmic and exponential functions as initial condition. A general framework of these methods is presented for analytical treatment of fractional partial differential equation arises in fluid mechanics. The fractional derivatives are described in the Caputo sense. The equation used in this paper is fractional wave equation, fractional burgers equation and fractional Klein-Gordon equation. After comparison of the results, the series of solution are found which is very helpful. The basic idea described in this paper is accepted to be further in use to solve other similar linear problems in fractional calculus.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM
AU  - Kirtiwant Parshuram Ghadle
AU  - Khan Firdous
AU  - Khan Arshiya Anjum
Y1  - 2017/01/13
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170201.13
DO  - 10.11648/j.ajmcm.20170201.13
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  - 13
EP  - 23
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170201.13
AB  - By handling the one dimensional partial differential equation with three methods i.e. Adomain decomposition method(ADM), Variation iteration method(VIM) and the New iterative method(NIM) and applied logarithmic and exponential functions as initial condition. A general framework of these methods is presented for analytical treatment of fractional partial differential equation arises in fluid mechanics. The fractional derivatives are described in the Caputo sense. The equation used in this paper is fractional wave equation, fractional burgers equation and fractional Klein-Gordon equation. After comparison of the results, the series of solution are found which is very helpful. The basic idea described in this paper is accepted to be further in use to solve other similar linear problems in fractional calculus.
VL  - 2
IS  - 1
ER  -

Copy | Download