Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term
A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2016/1015634 |
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| _version_ | 1832552785322704896 |
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| author | Essam R. El-Zahar |
| author_facet | Essam R. El-Zahar |
| author_sort | Essam R. El-Zahar |
| collection | DOAJ |
| description | A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term. |
| format | Article |
| id | doaj-art-82a0be1809764525bc5c73495dbbd9d3 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-82a0be1809764525bc5c73495dbbd9d32025-02-03T05:57:45ZengWileyInternational Journal of Differential Equations1687-96431687-96512016-01-01201610.1155/2016/10156341015634Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source TermEssam R. El-Zahar0Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi ArabiaA reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.http://dx.doi.org/10.1155/2016/1015634 |
| spellingShingle | Essam R. El-Zahar Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term International Journal of Differential Equations |
| title | Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term |
| title_full | Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term |
| title_fullStr | Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term |
| title_full_unstemmed | Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term |
| title_short | Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term |
| title_sort | piecewise approximate analytical solutions of high order singular perturbation problems with a discontinuous source term |
| url | http://dx.doi.org/10.1155/2016/1015634 |
| work_keys_str_mv | AT essamrelzahar piecewiseapproximateanalyticalsolutionsofhighordersingularperturbationproblemswithadiscontinuoussourceterm |