Controlling the dynamics of Burgers equation with a high-order nonlinearity
We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e., ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball in L2[0,1] and uniqueness of steady state solutions for all integer n≥1. Then, we use an adaptive nonlinear boundary controller to s...
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| Format: | Article |
| Language: | English |
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Wiley
2004-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171204404116 |
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| _version_ | 1832550734356283392 |
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| author | Nejib Smaoui |
| author_facet | Nejib Smaoui |
| author_sort | Nejib Smaoui |
| collection | DOAJ |
| description | We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e.,
ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball in L2[0,1]
and uniqueness of steady state solutions for all integer n≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results. |
| format | Article |
| id | doaj-art-e30fb7c8276f4ef48ffc940997430e48 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2004-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e30fb7c8276f4ef48ffc940997430e482025-02-03T06:06:10ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252004-01-012004623321333210.1155/S0161171204404116Controlling the dynamics of Burgers equation with a high-order nonlinearityNejib Smaoui0Department of Mathematics & Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, KuwaitWe investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e., ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball in L2[0,1] and uniqueness of steady state solutions for all integer n≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results.http://dx.doi.org/10.1155/S0161171204404116 |
| spellingShingle | Nejib Smaoui Controlling the dynamics of Burgers equation with a high-order nonlinearity International Journal of Mathematics and Mathematical Sciences |
| title | Controlling the dynamics of Burgers equation with a high-order nonlinearity |
| title_full | Controlling the dynamics of Burgers equation with a high-order nonlinearity |
| title_fullStr | Controlling the dynamics of Burgers equation with a high-order nonlinearity |
| title_full_unstemmed | Controlling the dynamics of Burgers equation with a high-order nonlinearity |
| title_short | Controlling the dynamics of Burgers equation with a high-order nonlinearity |
| title_sort | controlling the dynamics of burgers equation with a high order nonlinearity |
| url | http://dx.doi.org/10.1155/S0161171204404116 |
| work_keys_str_mv | AT nejibsmaoui controllingthedynamicsofburgersequationwithahighordernonlinearity |