An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry
We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is bas...
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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/10.1155/2014/425648 |
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| author | A. H. Bhrawy A. S. Alofi R. A. Van Gorder |
| author_facet | A. H. Bhrawy A. S. Alofi R. A. Van Gorder |
| author_sort | A. H. Bhrawy |
| collection | DOAJ |
| description | We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials Jn(α,β)(r) with α,β∈(-1,∞), r∈(0,1) and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity. |
| format | Article |
| id | doaj-art-4fc6209ffd334e06b957bb5ba2734ac2 |
| 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-4fc6209ffd334e06b957bb5ba2734ac22025-02-03T05:59:25ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/425648425648An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and GeometryA. H. Bhrawy0A. S. Alofi1R. A. Van Gorder2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, University of Central Florida, Orlando, FL 32816, USAWe present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials Jn(α,β)(r) with α,β∈(-1,∞), r∈(0,1) and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity.http://dx.doi.org/10.1155/2014/425648 |
| spellingShingle | A. H. Bhrawy A. S. Alofi R. A. Van Gorder An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry Abstract and Applied Analysis |
| title | An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry |
| title_full | An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry |
| title_fullStr | An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry |
| title_full_unstemmed | An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry |
| title_short | An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry |
| title_sort | efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry |
| url | http://dx.doi.org/10.1155/2014/425648 |
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