Authors:
Mankirat Kaur,Rantej Sharma,Kashish Wadhawan,Abhinav Dhiman,DOI NO:
https://doi.org/10.26782/jmcms.spl.11/2024.05.00016Keywords:
Boundary Conditions,Error Analysis,He’s Polynomial,Heat Equations,Nonlinear Terms,Homotopy ,Perturbation Method,Abstract
In this paper, we are examining He’s polynomial method for solving (1+1)- dimensional and (2+1)-dimensional heat-like equations that arise in various diffusion processes. The absolute error is calculated from the exact solution and numerical solution by taking different iterations of the He’s polynomial. This method is also called the homotopy perturbation method (HPM). The nonlinear terms can be easily handled by the use of He’s polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids round-off errors. Some examples are given to show the efficiency and accuracy of the He’s polynomial used to solve Heat-like equations.Refference:
I. Ghorbani. : ‘Beyond Adomian polynomials: He polynomials’. Chaos, Solitons & Fractals. vol. 39(3), pp. 1486–1492, 2009. 10.1016/j.chaos.2007.06.034
II. Ghorbani and J. Saberi-Nadjafi. : ‘He’s homotopy perturbation method for calculating adomian polynomials’. International Journal of Nonlinear Sciences and Numerical Simulation. vol. 8(2), pp. 229–232, 2007. 10.1515/IJNSNS.2007.8.2.229
III. J.-H. He. : ‘A coupling method of a homotopy technique and a perturbation technique for nonlinear problems’. International Journal of Non-Linear Mechanics. vol. 35(1), pp. 37–43, 2000. 10.1016/S0020-
7462(98)00085-7
IV. J.-H. He. : ‘An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering’. International Journal of Modern Physics B. vol. 22(21), pp. 3487–3578, 2008. 10.1142/S0217979208048668
V. J.-H. He. : ‘Comparison of homotopy perturbation method and homotopy analysis method’. Applied Mathematics and Computation. vol. 156(2),
pp. 527–539, 2004. 10.1016/j.amc.2003.08.008
VI. J.-H. He. : ‘Homotopy perturbation method for bifurcation of nonlinear problems’. International Journal of Nonlinear Sciences and Numerical Simulation. vol. 6(2), pp. 207–208, 2005. 10.1515/IJNSNS.2005.6.2.207
VII. J.-H. He. : ‘Homotopy perturbation method for solving boundary value problems’. Physics Letters A. vol. 350(1-2), pp. 87–88, 2006. 10.1016/j.physleta.2005.10.005
VIII. J.-H. He. : ‘Recent development of the homotopy perturbation method’, Topological Methods in Nonlinear Analysis’. vol. 31(2), pp. 205–209, 2008.
IX. J.-H. He. : ‘Some asymptotic methods for strongly nonlinear equations’. International Journal of Modern Physics B’. vol. 20(10) pp. 1141–1199, 2006. 10.1142/S0217979206033796
X. J.-H. He. : ‘The homotopy perturbation method for nonlinear oscillators with discontinuities’. Applied Mathematics and Computation. vol. 151(1), pp. 287–292, 2004. 10.1016/S0096-3003(03)00341-2
XI. L. Xu. : ‘He’s homotopy perturbation method for a boundary layer equation in unbounded domain’. Computers & Mathematics with Applications. vol. 54(7-8), pp. 1067–1070, 2007.
10.1016/j.camwa.2006.12.052
XII. M. A. Noor and S. T. Mohyud-Din. : ‘Homotopy perturbation method for solving nonlinear higher order boundary value problems’. International Journal of Nonlinear Sciences and Numerical Simulation. vol. 9(4),
pp. 395–408, 2008. 10.1515/IJNSNS.2008.9.4.395
XIII. M. A. Noor and S. T. Mohyud-Din. : ‘Modified variational iteration method for heat and wave-like equations’. Acta Applicandae Mathematicae. vol. 104(3), pp. 257–269, 2008. 10.1007/s10440-008- 9255-x
XIV. M. A. Noor and S. T. Mohyud-Din. : ‘Modified variation alliteration method for heat and wave-like equations’. Acta Applicandae Mathematicae. vol. 104(3), pp. 257–269, 2008. 10.1007/s10440-008- 9255-x
XV. M. Wazwaz and A. Gorguis. : ‘Exact solutions for heat-like and wave- like equations with variable coefficients’. Applied Mathematics and Computation. vol. 149(1), pp. 15–29, 2004. 10.1016/S0096-
3003(02)00946-3
XVI. R. Wilcox. : ‘Closed-form solution of the differential equation (∂^2/∂x∂y+ax(∂/∂x)+by(∂/∂y)+cxy+∂/∂t)P(x,y,t) = 0 by normal-ordering exponential operators’. Journal of Mathematical Physics. vol. 11(4),
pp. 1235–1237, 1970. 10.1063/1.1665252
XVII. S. T. Mohyud-Din, M. A. Noor, and K. I. Noor. : ‘Traveling wave solutions of seventh-order generalized KdV equations using He’s polynomials’. International Journal of Nonlinear Sciences and Numerical Simulation. vol. 10(2), pp. 227–233, 2009.
10.1515/IJNSNS.2009.10.2.227
XVIII. S. T. Mohyud-Din and M. A. Noor. : ‘Homotopy perturbation method for solving fourth-order boundary value problems’. Mathematical Problems in Engineering’. vol. 2007, Article ID 98602, 15 pages, 2007.
10.1155/2007/98602
XIX. S. T. Mohyud-Din, M. A. Noor, and K. I. Noor. : ‘Homotopy perturbation method for unsteady flow of gas through a porous medium’. International Journal of Modern Physics B. In press.
XX. S. T. Mohyud-Din, M. A. Noor, and K. I. Noor. : ‘On the coupling of polynomials with correction functional’. International Journal of Modern Physicss B. In press.