Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative
The Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we ai...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2020/1274251 |
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| author | Xin Liu Kamran Yukun Yao |
| author_facet | Xin Liu Kamran Yukun Yao |
| author_sort | Xin Liu |
| collection | DOAJ |
| description | The Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we aim to approximate the solution of a fractional Riccati equation of order 0<β<1 with Atangana–Baleanu derivative (ABC). Our numerical scheme is based on Laplace transform (LT) and quadrature rule. We apply LT to the given fractional differential equation, which reduces it to an algebraic equation. The reduced equation is solved for the unknown in LT space. The solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. The Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme. |
| format | Article |
| id | doaj-art-d68377dfd569402ca0de4e3592b4d6d7 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-d68377dfd569402ca0de4e3592b4d6d72025-02-03T01:00:30ZengWileyJournal of Mathematics2314-46292314-47852020-01-01202010.1155/2020/12742511274251Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional DerivativeXin Liu0Kamran1Yukun Yao2School of Hebei College of Traditional Chinese Medicine, Shijiazhuang 050200, Hebei, ChinaDepartment of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtoon Khwa, PakistanSchool of Hebei College of Traditional Chinese Medicine, Shijiazhuang 050200, Hebei, ChinaThe Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we aim to approximate the solution of a fractional Riccati equation of order 0<β<1 with Atangana–Baleanu derivative (ABC). Our numerical scheme is based on Laplace transform (LT) and quadrature rule. We apply LT to the given fractional differential equation, which reduces it to an algebraic equation. The reduced equation is solved for the unknown in LT space. The solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. The Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme.http://dx.doi.org/10.1155/2020/1274251 |
| spellingShingle | Xin Liu Kamran Yukun Yao Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative Journal of Mathematics |
| title | Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative |
| title_full | Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative |
| title_fullStr | Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative |
| title_full_unstemmed | Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative |
| title_short | Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative |
| title_sort | numerical approximation of riccati fractional differential equation in the sense of caputo type fractional derivative |
| url | http://dx.doi.org/10.1155/2020/1274251 |
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