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Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense

DOI: 10.4236/ajcm.2016.62008, PP. 66-73

Keywords: Random Partial Differential Equations (RPDEs), Mean Square Sense (m.s), Second Order Random Variable (2r.v.&apos,s), Random Crank-Nicolson Scheme, Convergence, Consistency, Stability

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Abstract:

The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.

References

[1]  Soong, T.T. (1973) Random Differential Equations in Science and Engineering. Academic Press, New York.
[2]  Thomas, J. (1998) Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics. Springer.
[3]  Allen, E.J., Novosel, S.J. and Zhang, Z.C. (1998) Finite Element and Difference Approximation of Some Linear Stochastic Partial Differential Equations. Stochastics and Stochastic Reports, 64, 117-142.
http://dx.doi.org/10.1080/17442509808834159
[4]  Davie, A.J. and Gaines, J.G. (2001) Convergence of Numerical Schemes for the Solution of Parabolic Stochastic Partial Differential Equations. Mathematics of Computations, 70, 121-134.
http://dx.doi.org/10.1090/S0025-5718-00-01224-2
[5]  McDonald, S. (2006) Finite Difference Approximation for Linear Stochastic Partial Differential Equation with Method of Lines. MPRA Paper, 3983.
[6]  Cortes, J.C., Lucas Jódar, L. and Villafuerte, R.J. (2007) Villanueva: Computing Mean Square Approximations of Random Diffusion Models with Source Term. Mathematics and Computers in Simulation, 76, 44-48.
http://dx.doi.org/10.1016/j.matcom.2007.01.020
[7]  El-Tawil, M.A. and Sohaly, M.A. (2009) Mean Square Numerical Methods for Initial Value Random Differential Equations. Open Journal of Discrete Mathematics (OJDM), 1, 66-84,
[8]  El-Tawil, M.A. and Sohaly, M.A. (2012) Mean Square Convergent Three Points Finite Difference Scheme for Random Partial Differential Equations. Journal of the Egyptian Mathematical Society, 20, 188-204.
http://dx.doi.org/10.1016/j.joems.2012.08.017
[9]  Sohaly, M.A. (2014) Mean Square Heun’s Method Convergent for Solving Random Differential Initial Value Problems of First Order. American Journal of Computational Mathematics, 4, 280-288.
[10]  Mohammed, A.S. (2014) Mean Square Convergent Three and Five Points Finite Difference Scheme for Stochastic Parabolic Partial Differential Equations. Electronic Journal of Mathematical Analysis and Applications, 2, 164-171.
[11]  Mohammed, W., Sohaly, M.A., El-Bassiouny, A. and Elnagar, K. (2014) Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs. American Journal of Computational Mathematics, 4, 280-288.
http://dx.doi.org/10.4236/ajcm.2014.44024

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