Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations
In this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Capu...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-05-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/6/6/288 |
| _version_ | 1797487337046802432 |
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| author | Hakeem Ullah Mehreen Fiza Ilyas Khan Nawa Alshammari Nawaf N. Hamadneh Saeed Islam |
| author_facet | Hakeem Ullah Mehreen Fiza Ilyas Khan Nawa Alshammari Nawaf N. Hamadneh Saeed Islam |
| author_sort | Hakeem Ullah |
| collection | DOAJ |
| description | In this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Caputo sense. A rapidly convergent series solution is obtained from the FOAFM and is validated by comparing with other results. The analysis proves that our method is simplified and applicable, contains less computational work, and has fast convergence. The beauty of this method is that there is no need to assume a small parameter such as in the perturbation method. The effectiveness and accuracy of the method is proven by numerical and graphical results. |
| first_indexed | 2024-03-09T23:47:10Z |
| format | Article |
| id | doaj.art-d59eab4ddc5241d79b7ac77a440cf1da |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T23:47:10Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-d59eab4ddc5241d79b7ac77a440cf1da2023-11-23T16:42:12ZengMDPI AGFractal and Fractional2504-31102022-05-016628810.3390/fractalfract6060288Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV EquationsHakeem Ullah0Mehreen Fiza1Ilyas Khan2Nawa Alshammari3Nawaf N. Hamadneh4Saeed Islam5Department of Mathematics, Abdul Wali Khan University, Mardan 23200, PakistanDepartment of Mathematics, Abdul Wali Khan University, Mardan 23200, PakistanDepartment of Mathematics, College of Science, Al-Zulfi Majmmah University, Al-Majmmah 11952, Saudi ArabiaDepartment of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi ArabiaDepartment of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi ArabiaDepartment of Mathematics, Abdul Wali Khan University, Mardan 23200, PakistanIn this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Caputo sense. A rapidly convergent series solution is obtained from the FOAFM and is validated by comparing with other results. The analysis proves that our method is simplified and applicable, contains less computational work, and has fast convergence. The beauty of this method is that there is no need to assume a small parameter such as in the perturbation method. The effectiveness and accuracy of the method is proven by numerical and graphical results.https://www.mdpi.com/2504-3110/6/6/288FOAFMclosed form solutionsKdV equationshomotopy analysis methodhomotopy perturbation method |
| spellingShingle | Hakeem Ullah Mehreen Fiza Ilyas Khan Nawa Alshammari Nawaf N. Hamadneh Saeed Islam Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations Fractal and Fractional FOAFM closed form solutions KdV equations homotopy analysis method homotopy perturbation method |
| title | Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations |
| title_full | Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations |
| title_fullStr | Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations |
| title_full_unstemmed | Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations |
| title_short | Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations |
| title_sort | modification of the optimal auxiliary function method for solving fractional order kdv equations |
| topic | FOAFM closed form solutions KdV equations homotopy analysis method homotopy perturbation method |
| url | https://www.mdpi.com/2504-3110/6/6/288 |
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