Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators
Laplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion t...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/6542787 |
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| author | null Kamran Ujala Gul Fahad M. Alotaibi Kamal Shah Thabet Abdeljawad |
| author_facet | null Kamran Ujala Gul Fahad M. Alotaibi Kamal Shah Thabet Abdeljawad |
| author_sort | null Kamran |
| collection | DOAJ |
| description | Laplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion techniques are then used to convert the obtained solution from Laplace domain into time domain. Various famous methods for numerical inversion of Laplace transform are based on quadrature approximation of Bromwich integral. The key features are the contour deformation and the choice of the quadrature rule. In this work, the Gauss–Hermite quadrature method and the contour integration method based on the trapezoidal and midpoint rule are tested and evaluated according to the criteria of applicability to actual inversion problems, applicability to different types of fractional differential equations, numerical accuracy, computational efficiency, and ease of programming and implementation. The performance and efficiency of the methods are demonstrated with the help of figures and tables. It is observed that the proposed methods converge rapidly with optimal accuracy without any time instability. |
| format | Article |
| id | doaj-art-82174d61845a4271904aeffa17c2ec65 |
| institution | OA Journals |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-82174d61845a4271904aeffa17c2ec652025-08-20T02:08:30ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/6542787Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operatorsnull Kamran0Ujala Gul1Fahad M. Alotaibi2Kamal Shah3Thabet Abdeljawad4Department of MathematicsDepartment of MathematicsDepartment of Information SystemsDepartment of Mathematics and SciencesDepartment of Mathematics and SciencesLaplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion techniques are then used to convert the obtained solution from Laplace domain into time domain. Various famous methods for numerical inversion of Laplace transform are based on quadrature approximation of Bromwich integral. The key features are the contour deformation and the choice of the quadrature rule. In this work, the Gauss–Hermite quadrature method and the contour integration method based on the trapezoidal and midpoint rule are tested and evaluated according to the criteria of applicability to actual inversion problems, applicability to different types of fractional differential equations, numerical accuracy, computational efficiency, and ease of programming and implementation. The performance and efficiency of the methods are demonstrated with the help of figures and tables. It is observed that the proposed methods converge rapidly with optimal accuracy without any time instability.http://dx.doi.org/10.1155/2023/6542787 |
| spellingShingle | null Kamran Ujala Gul Fahad M. Alotaibi Kamal Shah Thabet Abdeljawad Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators Journal of Mathematics |
| title | Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators |
| title_full | Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators |
| title_fullStr | Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators |
| title_full_unstemmed | Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators |
| title_short | Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators |
| title_sort | computational approach for differential equations with local and nonlocal fractional order differential operators |
| url | http://dx.doi.org/10.1155/2023/6542787 |
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