Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations
We will combine linear successive overrelaxation method with nonlinear monotone iterative scheme to obtain a new iterative method for solving nonlinear equations. The basic idea of this method joining traditional monotone iterative method (known as the method of lower and upper solutions) which depe...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/487273 |
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| _version_ | 1850237785288474624 |
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| author | Hung-Yu Ke Ren-Chuen Chen Chun-Hsien Li |
| author_facet | Hung-Yu Ke Ren-Chuen Chen Chun-Hsien Li |
| author_sort | Hung-Yu Ke |
| collection | DOAJ |
| description | We will combine linear successive overrelaxation method with nonlinear
monotone iterative scheme to obtain a new iterative method for solving nonlinear equations. The
basic idea of this method joining traditional monotone iterative method (known as the method
of lower and upper solutions) which depends essentially on the monotone parameter is that by
introducing an acceleration parameter one can construct a sequence to accelerate the convergence.
The resulting increase in the speed of convergence is very dramatic. Moreover, the sequence can
accomplish monotonic convergence behavior in the iterative process when some suitable acceleration
parameters are chosen. Under some suitable assumptions in aspect of the nonlinear function and
the matrix norm generated from this method, we can prove the boundedness and convergence of
the resulting sequences. Application of the iterative scheme is given to a logistic model problem
in ecology, and numerical results for a test problem with known analytical solution are given to
demonstrate the accuracy and efficiency of the present method. |
| format | Article |
| id | doaj-art-1d63f205ac284bfc975d34aeb9e68f21 |
| institution | OA Journals |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-1d63f205ac284bfc975d34aeb9e68f212025-08-20T02:01:40ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/487273487273Convergence Analysis of an Iterative Method for Nonlinear Partial Differential EquationsHung-Yu Ke0Ren-Chuen Chen1Chun-Hsien Li2Department of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, TaiwanDepartment of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, TaiwanDepartment of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, TaiwanWe will combine linear successive overrelaxation method with nonlinear monotone iterative scheme to obtain a new iterative method for solving nonlinear equations. The basic idea of this method joining traditional monotone iterative method (known as the method of lower and upper solutions) which depends essentially on the monotone parameter is that by introducing an acceleration parameter one can construct a sequence to accelerate the convergence. The resulting increase in the speed of convergence is very dramatic. Moreover, the sequence can accomplish monotonic convergence behavior in the iterative process when some suitable acceleration parameters are chosen. Under some suitable assumptions in aspect of the nonlinear function and the matrix norm generated from this method, we can prove the boundedness and convergence of the resulting sequences. Application of the iterative scheme is given to a logistic model problem in ecology, and numerical results for a test problem with known analytical solution are given to demonstrate the accuracy and efficiency of the present method.http://dx.doi.org/10.1155/2013/487273 |
| spellingShingle | Hung-Yu Ke Ren-Chuen Chen Chun-Hsien Li Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations Journal of Applied Mathematics |
| title | Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations |
| title_full | Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations |
| title_fullStr | Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations |
| title_full_unstemmed | Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations |
| title_short | Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations |
| title_sort | convergence analysis of an iterative method for nonlinear partial differential equations |
| url | http://dx.doi.org/10.1155/2013/487273 |
| work_keys_str_mv | AT hungyuke convergenceanalysisofaniterativemethodfornonlinearpartialdifferentialequations AT renchuenchen convergenceanalysisofaniterativemethodfornonlinearpartialdifferentialequations AT chunhsienli convergenceanalysisofaniterativemethodfornonlinearpartialdifferentialequations |