Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion
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 technique...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/843714 |
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| author | Mohamed A. El-Beltagy Noha A. Al-Mulla |
| author_facet | Mohamed A. El-Beltagy Noha A. Al-Mulla |
| author_sort | Mohamed A. El-Beltagy |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-825adc524f8c4253aa8c7f129ff1494b |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-825adc524f8c4253aa8c7f129ff1494b2025-02-03T01:26:13ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/843714843714Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite ExpansionMohamed A. El-Beltagy0Noha A. Al-Mulla1Electrical & Computer Engineering Department, Engineering Faculty, Effat University, Jeddah 21478, Saudi ArabiaMathematics Department, Science College, University of Dammam, Dammam 31451, Saudi ArabiaIn 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.http://dx.doi.org/10.1155/2014/843714 |
| spellingShingle | Mohamed A. El-Beltagy Noha A. Al-Mulla Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion Journal of Applied Mathematics |
| title | Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion |
| title_full | Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion |
| title_fullStr | Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion |
| title_full_unstemmed | Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion |
| title_short | Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion |
| title_sort | solution of the stochastic heat equation with nonlinear losses using wiener hermite expansion |
| url | http://dx.doi.org/10.1155/2014/843714 |
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