Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations
This paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series and vali...
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MDPI AG
2025-04-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/4/253 |
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| author | Fang Wang Qing Fang Yanyan Hu |
| author_facet | Fang Wang Qing Fang Yanyan Hu |
| author_sort | Fang Wang |
| collection | DOAJ |
| description | This paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series and validates its feasibility through comparison with known exact solutions. The proposed approach introduces a convergence parameter <i>ℏ</i>, which plays a crucial role in adjusting the convergence range of the series solution. By appropriately selecting initial terms, the convergence speed and computational accuracy can be significantly improved. The Jafari transform can be regarded as a generalization of classical transforms such as the Laplace and Elzaki transforms, enhancing the flexibility of the method. Numerical results demonstrate that the proposed technique is computationally efficient and easy to implement. Additionally, when the convergence parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℏ</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, both the homotopy perturbation method and the Adomian decomposition method emerge as special cases of the proposed method. The knowledge gained in this study will be important for model solving in the fields of mathematical economics, analysis of biological population dynamics, engineering optimization, and signal processing. |
| format | Article |
| id | doaj-art-bc44257c542e47b6b193cfbba7aec5f7 |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-bc44257c542e47b6b193cfbba7aec5f72025-08-20T02:28:14ZengMDPI AGFractal and Fractional2504-31102025-04-019425310.3390/fractalfract9040253Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential EquationsFang Wang0Qing Fang1Yanyan Hu2School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410001, ChinaSchool of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410001, ChinaSchool of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410001, ChinaThis paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series and validates its feasibility through comparison with known exact solutions. The proposed approach introduces a convergence parameter <i>ℏ</i>, which plays a crucial role in adjusting the convergence range of the series solution. By appropriately selecting initial terms, the convergence speed and computational accuracy can be significantly improved. The Jafari transform can be regarded as a generalization of classical transforms such as the Laplace and Elzaki transforms, enhancing the flexibility of the method. Numerical results demonstrate that the proposed technique is computationally efficient and easy to implement. Additionally, when the convergence parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℏ</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, both the homotopy perturbation method and the Adomian decomposition method emerge as special cases of the proposed method. The knowledge gained in this study will be important for model solving in the fields of mathematical economics, analysis of biological population dynamics, engineering optimization, and signal processing.https://www.mdpi.com/2504-3110/9/4/253system of partial differential equationshomotopy analysis methodCaputo fractional derivativesJafari transform methodfractional calculus |
| spellingShingle | Fang Wang Qing Fang Yanyan Hu Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations Fractal and Fractional system of partial differential equations homotopy analysis method Caputo fractional derivatives Jafari transform method fractional calculus |
| title | Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations |
| title_full | Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations |
| title_fullStr | Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations |
| title_full_unstemmed | Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations |
| title_short | Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations |
| title_sort | homotopy analysis transform method for solving systems of fractional order partial differential equations |
| topic | system of partial differential equations homotopy analysis method Caputo fractional derivatives Jafari transform method fractional calculus |
| url | https://www.mdpi.com/2504-3110/9/4/253 |
| work_keys_str_mv | AT fangwang homotopyanalysistransformmethodforsolvingsystemsoffractionalorderpartialdifferentialequations AT qingfang homotopyanalysistransformmethodforsolvingsystemsoffractionalorderpartialdifferentialequations AT yanyanhu homotopyanalysistransformmethodforsolvingsystemsoffractionalorderpartialdifferentialequations |