Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods

DOI: 10.4236/ojdm.2011.11002   PDF   HTML     4,537 Downloads   9,183 Views   Citations


In this paper, the cubic and quintic diffusion equation under stochastic non homogeneity is solved using Wiener- Hermite expansion and perturbation (WHEP) technique, Homotopy perturbation method (HPM) and Pickard approximation technique. The analytic solution of the linear case is obtained using Eigenfunction expansion .The Picard approximation method is used to introduce the first and second order approximate solution for the non linear case. The WHEP technique is also used to obtain approximate solution under different orders and different corrections. The Homotopy perturbation method (HPM) is also used to obtain some approximation orders for mean and variance. Using mathematica-5, the methods of solution are illustrated through figures, comparisons among different methods and some parametric studies.

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M. El-Tawil and A. Fareed, "Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods," Open Journal of Discrete Mathematics, Vol. 1 No. 1, 2011, pp. 6-21. doi: 10.4236/ojdm.2011.11002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Kampe de Feriet: Random solutions of partial differential equations, In: Proc. 3rd Berkeley Symposiumon Mathematical Statistics and Probability–1955, Vol. III, pp. 199- 208 (1956).
[2] Bharucha-Reid: A survey on the theory of random functions. The Institute of Mathematical Sciences: Matscience Report 31, India (1965).
[3] Lo Dato, V.: Stochastic processes in heat and mass transport, C. In: Bharucha-Reid (ed.) Probabilistic Methods in Applied Mathematics, Vol. 3, pp. 183-212. Academic, New York (1973).
[4] Becus, A. G.: Random generalized solutions to the heat equations. J. Math. Anal. Appl. 60, 93-102 (1977). doi:10.1016/0022-247X(77)90051-8
[5] Marcus, R.: Parabolic Ito equation with monotone nonlinearities. J. Funct. Anal. 29, 257-286 (1978). doi:10.1016/0022-1236(78)90031-9
[6] Manthey, R.: Weak convergence of solutions of the heat equation with Gaussian noise. Math. Nachr. 123, 157-168 (1985). doi:10.1002/mana.19851230115
[7] Manthey, R.: Existence and uniqueness of a solution of a reaction-diffusion with polynomial nonlinearity and with noise disturbance. Math. Nachr. 125, 121-133 (1986).
[8] Jetschke, G.: II. Most probable states of a nonlinear Brownian bridge. Forschungsergebnisse (Jena) N/86/20 (1986).
[9] Jetschke, G.: III. Tunneling in a bistable infinite- dimensional potential. Forschungsergebnisse (Jena) N/86/40 (1986).
[10] El-Tawil, M.: Nonhomogeneous boundary value problems. J. Math. Anal. Appl. 200, 53-65 (1996). doi:10.1006/jmaa.1996.0190
[11] Uemura, H.: Construction of the solution of 1-dim heat equation with white noise potential and its asymptotic behaviour. Stoch. Anal. Appl. 14(4), 487-506 (1996). doi:10.1080/07362999608809452
[12] El-Tawil, M.: The application ofWHEP technique on partial differential equations. Int. J. Differ. Equ. Appl. 7(3), 325-337 (2003).
[13] El-Tawil, M.: The homotopy Wiener-Hermite expansion and perturbation technique (WHEP). In: Transactions on Computational Science I. LNCS, Vol. 4750, pp. 159-180. Springer, New York (2008). doi:10.1007/978-3-540-79299-4_9
[14] Crow, S., Canavan, G.: Relationship between a Wiener- Hermite expansion and an energy cascade. J. Fluid Mech. 41(2), 387-403 (1970). doi:10.1017/S0022112070000654
[15] El-Tawil M. and Noha A. El-Molla, The approximate solution of a nonlinear diffusion equation using some techniques, a comparison study, International Journal of Nonlinear Sciences and numerical Simulation, 10(3), 687-698, 2009.
[16] El-Tawil M. and Noha A. El-Molla, Solving nonlinear diffusion equations without stochastic homogeneity using the homotopy perturbation method, J. of applied mathematics, pp. 281-299, 2009.
[17] Farlow, S. J.: Partial Differential Equations for Scientists and Engineers. Wiley, New York (1982).
[18] He, J. H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257-292(1999). doi:10.1016/S0045-7825(99)00018-3
[19] He, J. H.: A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int. J. Nonlinear Mech. 35, 37-43 (2000). doi:10.1016/S0020-7462(98)00085-7
[20] He, J. H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl.Math. Comput. 135, 73-79 (2003). doi:10.1016/S0096-3003(01)00312-5
[21] He, J. H.: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl.Math. Comput. 151, 287-292 (2004). doi:10.1016/S0096-3003(03)00341-2
[22] El-Tawil M., International Journal of Differential Equations and its Applications, Vol. 7, No. 3, pp 325-337, 2003.
[23] El-Tawil M., The Homotopy Wiener-Hermite expansion and perturbation technique (WHEP), Transactions on Computational Science (Springer), accepted.
[24] Farlow S. J., Partial differential equations for scientists and engineers, Wiley, N. Y., 1982.
[25] Crow S. and Canavan G., Relationship between a Wiener-Hermite expansion and an energy cascade, J. of fluid mechanics, 41(2), pp. 387-403 (1970). doi:10.1017/S0022112070000654
[26] Saffman P., Application of Wiener-Hermite expansion to the diffusion of a passive scalar in a homogeneous turbulent flow, Physics of fluids, 12(9), pp. 1786-1798(1969). doi:10.1063/1.1692743
[27] Kahan W. and Siegel A., Cameron-Martin-Wiener method in turbulence and in Burger’s model: General formulae and application to late decay, J. of fluid mechanics, 41(3), pp. 593-618 (1970).
[28] Chorin and Alexandre J., Gaussian fields and random flow, J. of fluid of mechanics, 63(1), pp. 21-32(1974).
[29] Eftimiu and Cornel, First-order Wiener-Hermite expansion in the electromagnetic scattering by conducting rough surfaces, Radio science, 23(5), pp. 769-779(1988).
[30] Jahedi A. and Ahmadi G., Application of Wiener-Hermite expansion to non-stationary random vibration of a Duffing oscillator, J. of applied mechanics, Transactions ASME, 50(2), pp. 436-442(1983). doi:10.1115/1.3167056
[31] Tamura Y. and Nakayama J., A formula on the Hermite expansion and its aoolication to a random boundary value problem, IEICE Transactions on electronics, E86-C(8), pp. 1743-1748 (2003).
[32] Kayanuma Y. and Noba K., Wiener-Hermite expansion formalism for the stochastic model of a driven quantum system, Chemical physics, 268(1-3), pp. 177-188(2001). doi:10.1016/S0301-0104(01)00305-6
[33] Gawad E. and El-Tawil M., General stochastic oscillatory systems, Applied Mathematical Modelling, 17(6), pp. 329-335(1993). doi:10.1016/0307-904X(93)90058-O
[34] Imamura T. Meecham W. and Siegel A., Symbolic calculus of the Wiener process and Wiener-Hermite functionals, J. math. Phys., 6(5), pp. 695-706(1983).

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