Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations
Fractional functional differential equations with delay (FDDEs) have recently played a significant role in modeling of many real areas of sciences such as physics, engineering, biology, medicine, and economics. FDDEs often cannot be solved analytically so the approximate and numerical methods should...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2015/510875 |
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| author | V. G. Pimenov A. S. Hendy |
| author_facet | V. G. Pimenov A. S. Hendy |
| author_sort | V. G. Pimenov |
| collection | DOAJ |
| description | Fractional functional differential equations with delay (FDDEs) have recently played a significant role in modeling of many real areas of sciences such as physics, engineering, biology, medicine, and economics. FDDEs often cannot be solved analytically so the approximate and numerical methods should be adapted to solve these types of equations. In this paper we consider a new method of backward differentiation formula- (BDF-) type for solving FDDEs. This approach is based on the interval approximation of the true solution using the Clenshaw and Curtis formula that is based on the truncated shifted Chebyshev polynomials. It is shown that the new approach can be reformulated in an equivalent way as a Runge-Kutta method and the Butcher tableau of this method is given. Estimation of local and global truncating errors is deduced and this leads to the proof of the convergence for the proposed method. Illustrative examples of FDDEs are included to demonstrate the validity and applicability of the proposed approach. |
| format | Article |
| id | doaj-art-d1e5ff21d35749b9aac0f85fd2c4dde2 |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-d1e5ff21d35749b9aac0f85fd2c4dde22025-08-20T02:19:43ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/510875510875Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev ApproximationsV. G. Pimenov0A. S. Hendy1Ural Federal University, Ulitsa Mira. 19. Yekaterinburg 620002, RussiaUral Federal University, Ulitsa Mira. 19. Yekaterinburg 620002, RussiaFractional functional differential equations with delay (FDDEs) have recently played a significant role in modeling of many real areas of sciences such as physics, engineering, biology, medicine, and economics. FDDEs often cannot be solved analytically so the approximate and numerical methods should be adapted to solve these types of equations. In this paper we consider a new method of backward differentiation formula- (BDF-) type for solving FDDEs. This approach is based on the interval approximation of the true solution using the Clenshaw and Curtis formula that is based on the truncated shifted Chebyshev polynomials. It is shown that the new approach can be reformulated in an equivalent way as a Runge-Kutta method and the Butcher tableau of this method is given. Estimation of local and global truncating errors is deduced and this leads to the proof of the convergence for the proposed method. Illustrative examples of FDDEs are included to demonstrate the validity and applicability of the proposed approach.http://dx.doi.org/10.1155/2015/510875 |
| spellingShingle | V. G. Pimenov A. S. Hendy Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations Abstract and Applied Analysis |
| title | Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations |
| title_full | Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations |
| title_fullStr | Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations |
| title_full_unstemmed | Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations |
| title_short | Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations |
| title_sort | numerical studies for fractional functional differential equations with delay based on bdf type shifted chebyshev approximations |
| url | http://dx.doi.org/10.1155/2015/510875 |
| work_keys_str_mv | AT vgpimenov numericalstudiesforfractionalfunctionaldifferentialequationswithdelaybasedonbdftypeshiftedchebyshevapproximations AT ashendy numericalstudiesforfractionalfunctionaldifferentialequationswithdelaybasedonbdftypeshiftedchebyshevapproximations |