Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet Method
In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ-Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ-Haar-wavelet operational matrix of fr...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2021-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/3084110 |
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| author | F. M. Alharbi A. M. Zidan Muhammad Naeem Rasool Shah Kamsing Nonlaopon |
| author_facet | F. M. Alharbi A. M. Zidan Muhammad Naeem Rasool Shah Kamsing Nonlaopon |
| author_sort | F. M. Alharbi |
| collection | DOAJ |
| description | In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ-Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ-Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ-fractional integral of the Haar-wavelet by utilizing the φ-fractional integral operator. We also extended our method to nonlinear φ-FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ-FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed. |
| format | Article |
| id | doaj-art-68751e6ea7104f098ab96ce7b1356e7b |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-68751e6ea7104f098ab96ce7b1356e7b2025-02-03T01:25:08ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/30841103084110Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet MethodF. M. Alharbi0A. M. Zidan1Muhammad Naeem2Rasool Shah3Kamsing Nonlaopon4Deanship of Common First Year, Umm Al-Qura University, Makkah, Saudi ArabiaDepartment of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi ArabiaDeanship of Common First Year, Umm Al-Qura University, Makkah, Saudi ArabiaDepartment of Mathematics, Abdul Wali khan university, Mardan 23200, PakistanDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandIn this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ-Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ-Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ-fractional integral of the Haar-wavelet by utilizing the φ-fractional integral operator. We also extended our method to nonlinear φ-FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ-FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.http://dx.doi.org/10.1155/2021/3084110 |
| spellingShingle | F. M. Alharbi A. M. Zidan Muhammad Naeem Rasool Shah Kamsing Nonlaopon Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet Method Journal of Function Spaces |
| title | Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet Method |
| title_full | Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet Method |
| title_fullStr | Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet Method |
| title_full_unstemmed | Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet Method |
| title_short | Numerical Investigation of Fractional-Order Differential Equations via φ-Haar-Wavelet Method |
| title_sort | numerical investigation of fractional order differential equations via φ haar wavelet method |
| url | http://dx.doi.org/10.1155/2021/3084110 |
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