In this paper, the exact solution of the fourth - order parabolic equations with variable coefficients is obtained by using a new homotopy perturbation method (NHPM), theoretical consideration are discussed. Finally, three examples are illustrated to show the validity and applicability of the proposed method.
| Published in | Pure and Applied Mathematics Journal (Volume 4, Issue 6) |
| DOI | 10.11648/j.pamj.20150406.13 |
| Page(s) | 242-247 |
| 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), 2015. Published by Science Publishing Group |
New Homotopy Perturbation Method (NHPM), Fourth - Order Parabolic Equations
| [1] | A. Q. M. Khaliq, E. H. Twizell, A family of second order methods for variable coefficient fourth order parabolic partial differential equations, Internat. J. Comput. Math. 23 (1987) 63–76. |
| [2] | D. J. Gorman, Free Vibrations Analysis of Beams and Shafts, Wiley, New York, 1975. |
| [3] | C. Andrade, S. McKee, High frequency A. D. I. methods for fourth order parabolic equations with variable coefficients, Internat. J. Comput. Appl. Math 3 (1977) 11–14. |
| [4] | S. D. Conte, A stable implicit difference approximation to a fourth order parabolic equation, J. ACM 4 (1957) 210–212. |
| [5] | S. D. Conte, W. C. Royster, Convergence of finite difference solution to a solution of the equation of a vibrating beam, Proc. Amer. Math. Soc. 7 (1956) 742–749. |
| [6] | D. J. Evans, A stable explicit method for the finite difference solution of fourth order parabolic partial differential equations, Comput. J. 8 (1965) 280–287. |
| [7] | A. Q. M. Khaliq and E. H. Twizell, A family of second - order methods for variable coefficient fourth - order parabolic partial differential equations, Int. J. Comput. Math 23 (1987), pp. 63–76. |
| [8] | D. J. Evans, W. S. Yousef, A note on solving the fourth order parabolic equation by the AGE method, Internat. J. Comput. Math. 40 (1991) 93–97. |
| [9] | A. M. Wazwaz, Analytic treatment for variable coefficient fourth - order parabolic partial differential equations, Appl. Math. Comput. 123 (2001), pp. 219–227. |
| [10] | A. M. Wazwaz, Exact solutions for variable coefficients fourth - order parabolic partial differential equations in higher - dimensional spaces, Appl. Mathe. Comput. 130 (2002), pp. 415–424. |
| [11] | J. Biazar and H. Ghazvini, He’s variational iteration method for fourth - orde parabolic equations, Comput. Math.Appl. 54 (2007), pp. 1047–1054. |
| [12] | Deniz Ağırseven Turgut Öziş, He's homotopy perturbation method for fourth - order parabolic equations,, Int. J. Comput. Math. Jon (2009). |
| [13] | J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73 - 79. |
| [14] | J. H. He, Comparison of Homotopy perturbation method and Homotopy analysis method, Applied Mathematics and Computation 156 (2004) 527 - 539. |
| [15] | J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation 151 (2004) 287 - 292. |
| [16] | J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation 6 (2005)207 - 208. |
| [17] | J. H. He, Some asymptotic method for strongly nonlinear equations, International Journal of Modern Physics 20 (2006) 1144 - 1199. |
| [18] | J. Biazar, M. Eslami, A new technique for non - linear two - dimensional wave equation, Scientia Iranica, Transactions B: Mechanical Engineering, 2013. Mechanics, 35 (1), pp.37 - 43 (2000). |
| [19] | S. T. Mohyud - Din, A. Yildirim, Homotopy perturbation method for advection problems, Nonlinear Science Letters A 1 (2010) 307 - 312. |
| [20] | A. Beléndez, T. Beléndez, A. Markuez, and C. Neipp, Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators, Chaos Solitons and Fractals (2006), doi: 10.1016/j.chaos. 2006. 09. 070. |
| [21] | A. Beléndez, T. Beléndez, C. Neipp, A. Hernandez, and M. L. Alvarez, Approximate solutions of a non linear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method, Chaos Solitons and Fractals (2007): 10.1016/j.chaos, 2007. 01. 089. |
| [22] | A. Belendez, A. Hernandez, T. Belendez, E. Fernandez, M. L. Alvarez, and C. Neip, Application of He’s homotopy perturbation method to the Duffing - harmonic oscillator, Int. J. Nonlinear Sci. Numer. Simul. 8 (1) (2007), pp. 79–88. |
| [23] | J. Biazar and M. Eslami, “A new homotopy perturbation method for solving systems of partial differential equations, ” Comput. Math Appl., in Press. |
| [24] | J. Biazar and M. Eslami, “A new method for solving the hyperbolic telegraph equation, ” Comput. Math. Model, in Press. |
| [25] | Mostafa Eslami1 and Jafar Biazar 2, analytical solution of the klein - gordon equation by a new Homotopy Perturbation Method Computational Mathematics and Modeling, Vol. 25, No. 1, March, 2014. |
| [26] | Mohammed ELbadri, A New Homotopy Perturbation Method for Solving Laplace Equation Advances in Theoretical and Applied Mathematics ISSN 0973 - 4554 Volume 8, Number 4 (2013), pp. 237 - 242. |
APA Style
Mohamed Elbadri, Tarig. M. Elzaki. (2015). New Modification of Homotopy Perturbation Method and the Fourth - Order Parabolic Equations with Variable Coefficients. Pure and Applied Mathematics Journal, 4(6), 242-247. https://doi.org/10.11648/j.pamj.20150406.13
ACS Style
Mohamed Elbadri; Tarig. M. Elzaki. New Modification of Homotopy Perturbation Method and the Fourth - Order Parabolic Equations with Variable Coefficients. Pure Appl. Math. J. 2015, 4(6), 242-247. doi: 10.11648/j.pamj.20150406.13
AMA Style
Mohamed Elbadri, Tarig. M. Elzaki. New Modification of Homotopy Perturbation Method and the Fourth - Order Parabolic Equations with Variable Coefficients. Pure Appl Math J. 2015;4(6):242-247. doi: 10.11648/j.pamj.20150406.13
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Download@article{10.11648/j.pamj.20150406.13,
author = {Mohamed Elbadri and Tarig. M. Elzaki},
title = {New Modification of Homotopy Perturbation Method and the Fourth - Order Parabolic Equations with Variable Coefficients},
journal = {Pure and Applied Mathematics Journal},
volume = {4},
number = {6},
pages = {242-247},
doi = {10.11648/j.pamj.20150406.13},
url = {https://doi.org/10.11648/j.pamj.20150406.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150406.13},
abstract = {In this paper, the exact solution of the fourth - order parabolic equations with variable coefficients is obtained by using a new homotopy perturbation method (NHPM), theoretical consideration are discussed. Finally, three examples are illustrated to show the validity and applicability of the proposed method.},
year = {2015}
}
TY - JOUR T1 - New Modification of Homotopy Perturbation Method and the Fourth - Order Parabolic Equations with Variable Coefficients AU - Mohamed Elbadri AU - Tarig. M. Elzaki Y1 - 2015/10/13 PY - 2015 N1 - https://doi.org/10.11648/j.pamj.20150406.13 DO - 10.11648/j.pamj.20150406.13 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 242 EP - 247 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20150406.13 AB - In this paper, the exact solution of the fourth - order parabolic equations with variable coefficients is obtained by using a new homotopy perturbation method (NHPM), theoretical consideration are discussed. Finally, three examples are illustrated to show the validity and applicability of the proposed method. VL - 4 IS - 6 ER -
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