Paper title:

Solution of Two-dimensional Parabolic Equation Subject to Non-local Boundary Conditions Using Homotopy Perturbation Method

Published in: Issue 1, (Vol. 6) / 2012
Publishing date: 2011-04-11
Pages: 64-68
Author(s): SHARMA Puskar Raj , METHI Giriraj
Abstract. Aim of the paper is to investigate solution of twodimensional linear parabolic partial differential equation with non-local boundary conditions using Homotopy Perturbation Method (HPM). This method is not only reliable in obtaining solution of such problems in series form with high accuracy but it also guarantees considerable saving of the calculation volume and time as compared to other methods. The application of the method has been illustrated through an example
Keywords: Homotopy Perturbation Method; Two Dimensional Parabolic Equation; Finite Difference Method; Finite Element Method; Adomain Decomposition Method

1.W.T.Ang, “A method of solution for the one-dimensional heat equation subject to a non-local condition”, SEA Bull. Math. Vol. 26, pp. 197-203, 2002.

2.J.R.Cannon, and J.Van der Hoek, “Diffusion subject to specification of mass”, J.Math.Anal. Appl. Vol .115, pp. 517- 529, 1986.

3.J.R.Cannon, “The one- dimensional heat equation, Encyclopedia of Mathematics and Its Applications”, Addison-Wesley, California, 1984.

4.J.R. Cannon, and J.Vander Hoek, “The one phase Stefen problem subject to specification of energy”, Journal of Mathematical Analysis and Applications, Vol.86, pp.281-291,1982.

5.J.R.Cannon, Y. Lin, and S.Wang,” An implicit finite difference scheme for the diffusion equation subject to mass specification”, Int. J. Eng. Sci. Vol. 28, pp. 573- 578, 1990.

6.W.A.Day, “Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories “, Quart. Appl. Math. Vol .40 , pp. 319–330, 1982.

7.W.A.Day, “A decreasing property of solutions of a parabolic equation with applications to thermoelasticity and other theories”, Quart. Appl. Math. Vol . 41, pp. 468–475 , 1983 .

8.G. Ekolin, “Finite difference methods for a non-local boundary value problem for the heat equation, BIT , Vol. 31,pp. 245–255 , 1991.

9.Y. Lin, S. Xu, and H.M.Yin,” Finite difference approximations for a class of nonlocal parabolic equations “, Int. J. Math. Math. Sci. Vol . 20, 147–164, 1997.

10. Y.Lin, “Parabolic partial differential equations subject to nonlocal boundary conditions”, Ph.D. dissertation, Department of Pure and Applied Mathematics, Washington State University, 1988.

11. Y.Liu, “Numerical solution of the heat equation with nonlocal boundary conditions “, J. Comput. Appl. Math. Vol .110, pp. 115–127 , 1999.

12. S.Wang , and Y.Lin, “A numerical method for the diffusion equation with nonlocal boundary specifications “, Int. J. Eng. Sci. Vol .28, pp. 543–546, 1990.

13. A.M.Wazwaz, “ A reliable modification of Adomian decomposition Method”, Appl. Math. Comput. Vol. 102, pp. 77–86, 1999.

14. M.Dehghan, “A finite difference method for a non-local boundary value problem for two dimensional heat equation” , Applied Math. and Comput. Vol .112, pp.133-142, 2000.

15. M.Dehghan , “Second order schemes for a boundary value problem with Neumann boundary conditions “, Applied Math. and Comput. Vol .138, pp.173-184, 2002.

16. J.H.He”Homotopy perturbation technique, computational methods,” Applied Mecha. and Engg. Vol .178, pp.257-262, 1999.

17. J.H.He, “A coupling method of Homotopy technique and a Perturbation technique for nonlinear problems,” Int.Journal of Nonlinear Mech., Vol .35, pp.37-43, 2000.

18. J.H.He, “A simple perturbation approach to Blasius equation,” Applied Math. and Comput. Vol . 140, pp. 217-222 , 2003.

19. J.H.He, “Application of Homotopy Perturbation Method to nonlinear wave equation,” Chaos Solito. &Fract., 26, pp. 295- 300, 2005. 20. J.H.He, “Homotopy Perturbation Method for solving boundary value problem,” Phy. Lett. A . Vol .350, pp. 87-88 , 2006.

21. J. Biazer, and H.Ghazvini, “He ‘s Homotopy Perturbation Method for solving systems of volterra integral equations ,”Chaos ,Solito.& Fract..Vol. 39 , pp. 770-777 ,2009.

22. M.S.H Chowdhary, and I. Hashim, “Solutions of time dependent Emden-Fowler type equations by Homotopy Perturbation Method,”Phy. Lett. A. Vol. 368, pp. 305-313, 2007.

23. S.Abbasbandy, “Application of He’s Homotopy Perturbation Method to functional integral equations,” Chaos Solito. &Fract.Vol. 31, pp.1243-1247, 2007.

24. J.Biazer, “He’s Homotopy Perturbation Method for Solving Helmholtz Equation,”Int. J. Contemp. Math. Sciences, Vol. 3, pp. 739 – 744, 2008.

25. A.M.Wazwaz,”A new algorithm for calculating Adomian polynomials for nonlinear operators, applied mathematics and computations”,vol-111, pp.53-69, 2000.

26. J.W.Batten “Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations”. Math.Comput., Vol.17, pp. 405– 413, 1963.

27. S.A.Belin, “Existence of solutions for one-dimensional wave equations with non-local conditions”. Electron. J. Differ. Eq. , Vol.76, pp. 18, 2001.

28. P.Shi, “Weak solution to evolution problem with a non-local constraint”. SIAM J. Anal. , Vol.24, pp. 46–58, 1993.

29. Y.S. Choi, and K.Y.Chan, “A parabolic equation with nonlocal boundary conditions arising from electrochemistry”. Nonlinear Analysis, Vol.18, pp. 317-331, 1992.

30. N.I. Kamynin, “A boundary value problem in the theory of the heat conduction with non-classical boundary condition”. USSR Comput.Math. Math. Phys., Vol.4, pp. 33–59,1964.

31. N.I.Yurchuk, “Mixed problem with an integral condition for certain parabolic equations”. Differ. Equ. Vol.22, pp. 1457– 1463, 1986.

32. L.S.Pulkina, “A non-local problem with integral conditions for hyperbolic equations”. Electron. J. Differ. Eq., Vol. 45, pp. 1–6, 1999.

33. L.Bougoffa, “Weak solution for hyperbolic equations with a non-local condition”. Aust.J. Math. Anal. Appl., Vol.2, pp. 1–7, 2005.

34. J.R.Cannon, “The solution of the heat equation subject to the specification of energy”. Quart. Appl. Math., Vol.21, pp. 155– 160, 1983.

35. B.Cahlon, D.M. Kulkarni, .and P. Shi, “Stepwise stability for the heat equation with a non-local constraint”. SIAM J. Numer. Anal., Vol. 32, pp. 571–593, 1995

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