[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).
|