Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations
In this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously...
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Wiley
2022-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2022/7087481 |
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| author | Bo Xu Sheng Zhang |
| author_facet | Bo Xu Sheng Zhang |
| author_sort | Bo Xu |
| collection | DOAJ |
| description | In this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously deforms to the class of local fractional IDE. As a concrete example, an explicit and exact Mittag–Leffler function solution of one special case of the local fractional IDE is obtained. In the process of solving, two initial solutions are selected for the iterative operation of local fractional HPM. One of the initial solutions has a critical condition of convergence and divergence related to the fractional order, and the other converges directly to the real solution. This paper reveals that whether the sequence of approximate solutions generated by the iteration of local fractional HPM can approach the real solution depends on the selection of the initial approximate solutions and sometimes also depends on the fractional order of the selected initial approximate solutions or the considered equations. |
| format | Article |
| id | doaj-art-4044f73665704bbb8693c4a8af56155e |
| institution | Kabale University |
| issn | 1687-9139 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-4044f73665704bbb8693c4a8af56155e2025-02-03T05:57:22ZengWileyAdvances in Mathematical Physics1687-91392022-01-01202210.1155/2022/7087481Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential EquationsBo Xu0Sheng Zhang1School of MathematicsSchool of Mathematical SciencesIn this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously deforms to the class of local fractional IDE. As a concrete example, an explicit and exact Mittag–Leffler function solution of one special case of the local fractional IDE is obtained. In the process of solving, two initial solutions are selected for the iterative operation of local fractional HPM. One of the initial solutions has a critical condition of convergence and divergence related to the fractional order, and the other converges directly to the real solution. This paper reveals that whether the sequence of approximate solutions generated by the iteration of local fractional HPM can approach the real solution depends on the selection of the initial approximate solutions and sometimes also depends on the fractional order of the selected initial approximate solutions or the considered equations.http://dx.doi.org/10.1155/2022/7087481 |
| spellingShingle | Bo Xu Sheng Zhang Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations Advances in Mathematical Physics |
| title | Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations |
| title_full | Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations |
| title_fullStr | Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations |
| title_full_unstemmed | Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations |
| title_short | Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations |
| title_sort | modified homotopy perturbation method and approximate solutions to a class of local fractional integrodifferential equations |
| url | http://dx.doi.org/10.1155/2022/7087481 |
| work_keys_str_mv | AT boxu modifiedhomotopyperturbationmethodandapproximatesolutionstoaclassoflocalfractionalintegrodifferentialequations AT shengzhang modifiedhomotopyperturbationmethodandapproximatesolutionstoaclassoflocalfractionalintegrodifferentialequations |