The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations
The stochastic Θ-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochastic Θ-method is convergent of order 1/2 in mean-square sense for such equations....
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/583930 |
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| _version_ | 1832555384300109824 |
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| author | Peng Hu Chengming Huang |
| author_facet | Peng Hu Chengming Huang |
| author_sort | Peng Hu |
| collection | DOAJ |
| description | The stochastic Θ-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochastic Θ-method is convergent of order 1/2 in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochastic Θ-method is mean-square asymptotically stable for every stepsize if 1/2≤θ≤1 and when 0≤θ<1/2, the stochastic Θ-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments. |
| format | Article |
| id | doaj-art-815ef690125d4469b1df816e280ef061 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-815ef690125d4469b1df816e280ef0612025-02-03T05:48:24ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/583930583930The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential EquationsPeng Hu0Chengming Huang1School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, ChinaSchool of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, ChinaThe stochastic Θ-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochastic Θ-method is convergent of order 1/2 in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochastic Θ-method is mean-square asymptotically stable for every stepsize if 1/2≤θ≤1 and when 0≤θ<1/2, the stochastic Θ-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.http://dx.doi.org/10.1155/2014/583930 |
| spellingShingle | Peng Hu Chengming Huang The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations Abstract and Applied Analysis |
| title | The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations |
| title_full | The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations |
| title_fullStr | The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations |
| title_full_unstemmed | The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations |
| title_short | The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations |
| title_sort | stochastic θ method for nonlinear stochastic volterra integro differential equations |
| url | http://dx.doi.org/10.1155/2014/583930 |
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