International Journal of Partial Differential Equations and Applications
ISSN (Print): 2376-9548 ISSN (Online): 2376-9556 Website: http://www.sciepub.com/journal/ijpdea Editor-in-chief: Mahammad Nurmammadov
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International Journal of Partial Differential Equations and Applications. 2015, 3(1), 20-24
DOI: 10.12691/ijpdea-3-1-4
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Approximate Solution of Stochastic Partial Differential Equation with Random Neumann Boundary Condition

A. H. EL_Bassiouny1, , W. W. Mohammed1 and F. Eskander1

1Department of Mathematics, Faculty of Science, Mansoura University, Egypt

Pub. Date: March 15, 2015

Cite this paper:
A. H. EL_Bassiouny, W. W. Mohammed and F. Eskander. Approximate Solution of Stochastic Partial Differential Equation with Random Neumann Boundary Condition. International Journal of Partial Differential Equations and Applications. 2015; 3(1):20-24. doi: 10.12691/ijpdea-3-1-4

Abstract

In this paper we approximate the solution of a parabolic nonlinear stochastic partial differential equation (SPDE) with cubic nonlinearity and with random Neumann boundary condition via a stochastic ordinary differential equation (SODE) which is a stochastic amplitude equation near a change of stability.

Keywords:
amplitude equations SPDEs random boundary conditions multiscale analysis Ginzburg-Landau equation.

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References:

[1]  D. Blomker and W. W. Mohammed. Amplitude equations for SPDEs with cubic nonlinearities. Stochastics. An International Journal of Probability and Stochastic Processes. 85 (2): 181-215, (2013).
 
[2]  S. Cerrai and M. Freidlin. Fast transport asymptotics for stochastic RDE.s with boundary noise, Ann. Probab. 39: 369.405, (2011).
 
[3]  G. Da Prato, S. Kwapien, and J. Zabczyk. Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics, 23:1-23, (1987).
 
[4]  G. Da Prato and J. Zabczyk. Evolution equations with white-noise boundary conditions. Stochastics Rep. 42: 167-182, (1993).
 
[5]  G. Da Prato and J. Zabczyk. Ergodicity for In.nite Dimensional Systems, volume 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, (1996).
 
[6]  R. E. Langer. A problem in diffusion or in the. ow of heat for a solid in contact with afuid. Tohoku Math. J.35: 260-275, (1932).
 
[7]  L. Lapidus and N. Amundson. Chemical reactor theory, Prentice-Hall, (1977).
 
[8]  J. P. Peixoto and A. H. Oort. Physics of climate. Springer, New York, (1992).
 
[9]  R. B. Sowers. Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab. 22: 2071.2121, (1994).
 
[10]  S. Xu, J. Duan. A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions. Applied Mathematics and Computation. 217, 9532-9542, (2011).