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Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener‐Hermite Expansion

Author

Listed:
  • Mohamed A. El-Beltagy
  • Noha A. Al-Mulla

Abstract

In the current work, the Wiener‐Hermite expansion (WHE) is used to solve the stochastic heat equation with nonlinear losses. WHE is used to deduce the equivalent deterministic system up to third order accuracy. The solution of the equivalent deterministic system is obtained using different techniques numerically and analytically. The finite‐volume method (FVM) with Pickard iteration is used to solve the equivalent system iteratively. The WHE with perturbation technique (WHEP) is applied to deduce more simple and decoupled equivalent deterministic system that can be solved numerically without iterations. The system resulting from WHEP technique is solved also analytically using the eigenfunction expansion technique. The Monte‐Carlo simulations (MCS) are performed to get the statistical properties of the stochastic solution and to verify other solution techniques. The results show that higher‐order solutions are essential especially in case of nonlinearities where non‐Gaussian effects cannot be neglected. The comparisons show the efficiency of the numerical WHE and WHEP techniques in solving stochastic nonlinear PDEs compared with the analytical solution and MCS.

Suggested Citation

  • Mohamed A. El-Beltagy & Noha A. Al-Mulla, 2014. "Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener‐Hermite Expansion," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:843714
    DOI: 10.1155/2014/843714
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    References listed on IDEAS

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    1. Mohamed A. El-Beltagy & Amnah S. Al-Johani, 2013. "Numerical Approximation of Higher‐Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    2. Mohamed A. El-Beltagy & Amnah S. Al-Johani, 2013. "Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-21, August.
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    Cited by:

    1. M.-C. Casabán & J.-C. Cortés & J.-V. Romero & M.-D. Roselló, 2016. "Random First‐Order Linear Discrete Models and Their Probabilistic Solution: A Comprehensive Study," Abstract and Applied Analysis, John Wiley & Sons, vol. 2016(1).

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