On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method
Differential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equatio...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/6/1214 |
| _version_ | 1797592549671567360 |
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| author | Kamran Sharif Ullah Khan Salma Haque Nabil Mlaiki |
| author_facet | Kamran Sharif Ullah Khan Salma Haque Nabil Mlaiki |
| author_sort | Kamran |
| collection | DOAJ |
| description | Differential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equations of fractional order is developed. The analytic inversion can be very difficult for complex forms of the transform function. Therefore, numerical methods are used for the inversion of the Laplace transform. In general, the numerical inverse Laplace transform is an ill-posed problem. This difficulty has led to various numerical methods for the inversion of the Laplace transform. In this work, the Weeks method is utilized for the numerical inversion of the Laplace transform. In our proposed numerical method, first, the fractional-order differential equation is converted to an algebraic equation using Laplace transform. Then, the transformed equation is solved in Laplace space using algebraic techniques. Finally, the Weeks method is utilized for the inversion of the Laplace transform. Weeks method is one of the most efficient numerical methods for the computation of the inverse Laplace transform. We have considered five test problems for validation of the proposed numerical method. Based on the comparison between analytical results and the Weeks method results, the reliability and effectiveness of the Weeks method for fractional-order differential equations was confirmed. |
| first_indexed | 2024-03-11T01:52:47Z |
| format | Article |
| id | doaj.art-15a42e0aaa0f4caf8383b48e694ad3e4 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T01:52:47Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-15a42e0aaa0f4caf8383b48e694ad3e42023-11-18T12:51:02ZengMDPI AGSymmetry2073-89942023-06-01156121410.3390/sym15061214On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks MethodKamran0Sharif Ullah Khan1Salma Haque2Nabil Mlaiki3Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, PakistanDepartment of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaDepartment of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi ArabiaDifferential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equations of fractional order is developed. The analytic inversion can be very difficult for complex forms of the transform function. Therefore, numerical methods are used for the inversion of the Laplace transform. In general, the numerical inverse Laplace transform is an ill-posed problem. This difficulty has led to various numerical methods for the inversion of the Laplace transform. In this work, the Weeks method is utilized for the numerical inversion of the Laplace transform. In our proposed numerical method, first, the fractional-order differential equation is converted to an algebraic equation using Laplace transform. Then, the transformed equation is solved in Laplace space using algebraic techniques. Finally, the Weeks method is utilized for the inversion of the Laplace transform. Weeks method is one of the most efficient numerical methods for the computation of the inverse Laplace transform. We have considered five test problems for validation of the proposed numerical method. Based on the comparison between analytical results and the Weeks method results, the reliability and effectiveness of the Weeks method for fractional-order differential equations was confirmed.https://www.mdpi.com/2073-8994/15/6/1214Laplace transformtime-fractional differential equationsnumerical inversionWeeks methodLaguerre polynomials |
| spellingShingle | Kamran Sharif Ullah Khan Salma Haque Nabil Mlaiki On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method Symmetry Laplace transform time-fractional differential equations numerical inversion Weeks method Laguerre polynomials |
| title | On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method |
| title_full | On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method |
| title_fullStr | On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method |
| title_full_unstemmed | On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method |
| title_short | On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method |
| title_sort | on the approximation of fractional order differential equations using laplace transform and weeks method |
| topic | Laplace transform time-fractional differential equations numerical inversion Weeks method Laguerre polynomials |
| url | https://www.mdpi.com/2073-8994/15/6/1214 |
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