Numerical treatment for solving fractional Riccati differential equation
This paper presents an accurate numerical method for solving fractional Riccati differential equation (FRDE). The proposed method so called fractional Chebyshev finite difference method (FCheb-FDM). In this technique, we approximate FRDE with a finite dimensional problem. The method is based on the...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2013-04-01
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| Series: | Journal of the Egyptian Mathematical Society |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110256X12000521 |
| _version_ | 1818037053735043072 |
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| author | M.M. Khader |
| author_facet | M.M. Khader |
| author_sort | M.M. Khader |
| collection | DOAJ |
| description | This paper presents an accurate numerical method for solving fractional Riccati differential equation (FRDE). The proposed method so called fractional Chebyshev finite difference method (FCheb-FDM). In this technique, we approximate FRDE with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. By this method the given problem is reduced to a problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FRDE. Special attention is given to study the convergence analysis and estimate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique. |
| first_indexed | 2024-12-10T07:20:44Z |
| format | Article |
| id | doaj.art-bcdff7a5e5ad46668edb82ea3192c1d4 |
| institution | Directory Open Access Journal |
| issn | 1110-256X |
| language | English |
| last_indexed | 2024-12-10T07:20:44Z |
| publishDate | 2013-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of the Egyptian Mathematical Society |
| spelling | doaj.art-bcdff7a5e5ad46668edb82ea3192c1d42022-12-22T01:57:49ZengSpringerOpenJournal of the Egyptian Mathematical Society1110-256X2013-04-01211323710.1016/j.joems.2012.09.005Numerical treatment for solving fractional Riccati differential equationM.M. KhaderThis paper presents an accurate numerical method for solving fractional Riccati differential equation (FRDE). The proposed method so called fractional Chebyshev finite difference method (FCheb-FDM). In this technique, we approximate FRDE with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. By this method the given problem is reduced to a problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FRDE. Special attention is given to study the convergence analysis and estimate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.http://www.sciencedirect.com/science/article/pii/S1110256X12000521Fractional Riccati differential equationCaputo fractional derivativeChebyshev polynomials approximationFinite difference methodConvergence analysis |
| spellingShingle | M.M. Khader Numerical treatment for solving fractional Riccati differential equation Journal of the Egyptian Mathematical Society Fractional Riccati differential equation Caputo fractional derivative Chebyshev polynomials approximation Finite difference method Convergence analysis |
| title | Numerical treatment for solving fractional Riccati differential equation |
| title_full | Numerical treatment for solving fractional Riccati differential equation |
| title_fullStr | Numerical treatment for solving fractional Riccati differential equation |
| title_full_unstemmed | Numerical treatment for solving fractional Riccati differential equation |
| title_short | Numerical treatment for solving fractional Riccati differential equation |
| title_sort | numerical treatment for solving fractional riccati differential equation |
| topic | Fractional Riccati differential equation Caputo fractional derivative Chebyshev polynomials approximation Finite difference method Convergence analysis |
| url | http://www.sciencedirect.com/science/article/pii/S1110256X12000521 |
| work_keys_str_mv | AT mmkhader numericaltreatmentforsolvingfractionalriccatidifferentialequation |