A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods
This paper presents an innovative approach to solve $ \mathit{q} $-fractional partial differential equations through a combination of two semi-analytical techniques: The Residual Power Series Method (RPSM) and the Homotopy Analysis Method (HAM). Both methods are extended to obtain approximations for...
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| Format: | Article |
| Language: | English |
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AIMS Press
2024-11-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241596 |
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| author | Khalid K. Ali Mohamed S. Mohamed M. Maneea |
| author_facet | Khalid K. Ali Mohamed S. Mohamed M. Maneea |
| author_sort | Khalid K. Ali |
| collection | DOAJ |
| description | This paper presents an innovative approach to solve $ \mathit{q} $-fractional partial differential equations through a combination of two semi-analytical techniques: The Residual Power Series Method (RPSM) and the Homotopy Analysis Method (HAM). Both methods are extended to obtain approximations for $ \mathit{q} $-fractional partial differential equations ($ \mathit{q} $-FPDEs). These equations are significant in $ \mathit{q} $-calculus, which has gained attention due to its relevance in engineering applications, particularly in quantum mechanics. In this study, we solve linear and nonlinear $ \mathit{q} $-FPDEs and obtain the closed-form solutions, which confirm the validity of the utilized methods. The results are further illustrated through two-dimensional and three-dimensional graphs, thus highlighting the interaction between parameters, particularly the fractional parameter, the $ \mathit{q} $-calculus parameter, and time. |
| format | Article |
| id | doaj-art-fe867cd69acb4629aba398aa701ca19b |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-fe867cd69acb4629aba398aa701ca19b2025-01-23T07:53:24ZengAIMS PressAIMS Mathematics2473-69882024-11-01912334423346610.3934/math.20241596A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methodsKhalid K. Ali0Mohamed S. Mohamed1M. Maneea2Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, EgyptDepartment of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaFaculty of Engineering, MTI University, Cairo, EgyptThis paper presents an innovative approach to solve $ \mathit{q} $-fractional partial differential equations through a combination of two semi-analytical techniques: The Residual Power Series Method (RPSM) and the Homotopy Analysis Method (HAM). Both methods are extended to obtain approximations for $ \mathit{q} $-fractional partial differential equations ($ \mathit{q} $-FPDEs). These equations are significant in $ \mathit{q} $-calculus, which has gained attention due to its relevance in engineering applications, particularly in quantum mechanics. In this study, we solve linear and nonlinear $ \mathit{q} $-FPDEs and obtain the closed-form solutions, which confirm the validity of the utilized methods. The results are further illustrated through two-dimensional and three-dimensional graphs, thus highlighting the interaction between parameters, particularly the fractional parameter, the $ \mathit{q} $-calculus parameter, and time.https://www.aimspress.com/article/doi/10.3934/math.20241596$ \mathit{q} $-calculusfractional calculus$ \mathit{q} $-fractional partial differential equationshomotopy analysis methodresidual power series methodsemi analytical techniques |
| spellingShingle | Khalid K. Ali Mohamed S. Mohamed M. Maneea A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods AIMS Mathematics $ \mathit{q} $-calculus fractional calculus $ \mathit{q} $-fractional partial differential equations homotopy analysis method residual power series method semi analytical techniques |
| title | A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods |
| title_full | A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods |
| title_fullStr | A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods |
| title_full_unstemmed | A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods |
| title_short | A novel approach to $ \mathit{q} $-fractional partial differential equations: Unraveling solutions through semi-analytical methods |
| title_sort | novel approach to mathit q fractional partial differential equations unraveling solutions through semi analytical methods |
| topic | $ \mathit{q} $-calculus fractional calculus $ \mathit{q} $-fractional partial differential equations homotopy analysis method residual power series method semi analytical techniques |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241596 |
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