The well-posedness and solutions of Boussinesq-type equations
We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive su...
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| Format: | Thesis |
| Language: | English |
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Curtin University
2009
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| Online Access: | http://hdl.handle.net/20.500.11937/2247 |
| _version_ | 1848743900982804480 |
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| author | Lin, Qun |
| author_facet | Lin, Qun |
| author_sort | Lin, Qun |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time.Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations.Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior. |
| first_indexed | 2025-11-14T05:52:55Z |
| format | Thesis |
| id | curtin-20.500.11937-2247 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T05:52:55Z |
| publishDate | 2009 |
| publisher | Curtin University |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-22472017-02-20T06:37:03Z The well-posedness and solutions of Boussinesq-type equations Lin, Qun analytical solution techniques well-posedness theory numerical solution techniques Jacobi/exponential expansion method Cauchy problem Boussinesq-type equations We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time.Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations.Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior. 2009 Thesis http://hdl.handle.net/20.500.11937/2247 en Curtin University fulltext |
| spellingShingle | analytical solution techniques well-posedness theory numerical solution techniques Jacobi/exponential expansion method Cauchy problem Boussinesq-type equations Lin, Qun The well-posedness and solutions of Boussinesq-type equations |
| title | The well-posedness and solutions of Boussinesq-type equations |
| title_full | The well-posedness and solutions of Boussinesq-type equations |
| title_fullStr | The well-posedness and solutions of Boussinesq-type equations |
| title_full_unstemmed | The well-posedness and solutions of Boussinesq-type equations |
| title_short | The well-posedness and solutions of Boussinesq-type equations |
| title_sort | well-posedness and solutions of boussinesq-type equations |
| topic | analytical solution techniques well-posedness theory numerical solution techniques Jacobi/exponential expansion method Cauchy problem Boussinesq-type equations |
| url | http://hdl.handle.net/20.500.11937/2247 |