A Numerical Solution for Hirota-Satsuma Coupled KdV Equation
A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubic B-splines as test functions and a linear B-spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newt...
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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/10.1155/2014/819367 |
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| author | M. S. Ismail H. A. Ashi |
| author_facet | M. S. Ismail H. A. Ashi |
| author_sort | M. S. Ismail |
| collection | DOAJ |
| description | A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubic B-splines as test functions and a linear B-spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newton’s method is used to solve the block nonlinear pentadiagonal system we have obtained. The resulting schemes are of second order accuracy in both directions, space and time. The von Neumann stability analysis of the schemes shows that the two schemes are unconditionally stable. The single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitons, three solitons, and birth of solitons is also discussed. |
| format | Article |
| id | doaj-art-a90754efec7b4be6b61277e5947f6ef5 |
| 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-a90754efec7b4be6b61277e5947f6ef52025-02-03T07:24:50ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/819367819367A Numerical Solution for Hirota-Satsuma Coupled KdV EquationM. S. Ismail0H. A. Ashi1Department of Mathematics, College of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, College of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaA Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubic B-splines as test functions and a linear B-spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newton’s method is used to solve the block nonlinear pentadiagonal system we have obtained. The resulting schemes are of second order accuracy in both directions, space and time. The von Neumann stability analysis of the schemes shows that the two schemes are unconditionally stable. The single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitons, three solitons, and birth of solitons is also discussed.http://dx.doi.org/10.1155/2014/819367 |
| spellingShingle | M. S. Ismail H. A. Ashi A Numerical Solution for Hirota-Satsuma Coupled KdV Equation Abstract and Applied Analysis |
| title | A Numerical Solution for Hirota-Satsuma Coupled KdV Equation |
| title_full | A Numerical Solution for Hirota-Satsuma Coupled KdV Equation |
| title_fullStr | A Numerical Solution for Hirota-Satsuma Coupled KdV Equation |
| title_full_unstemmed | A Numerical Solution for Hirota-Satsuma Coupled KdV Equation |
| title_short | A Numerical Solution for Hirota-Satsuma Coupled KdV Equation |
| title_sort | numerical solution for hirota satsuma coupled kdv equation |
| url | http://dx.doi.org/10.1155/2014/819367 |
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