Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM
This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integr...
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| Format: | Article |
| Language: | English |
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Vilnius Gediminas Technical University
2025-01-01
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| Series: | Mathematical Modelling and Analysis |
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| Online Access: | https://gc.vgtu.lt/index.php/MMA/article/view/20268 |
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| author | Zainidin Eshkuvatov Zaid Laadjal Shahrina Ismail |
| author_facet | Zainidin Eshkuvatov Zaid Laadjal Shahrina Ismail |
| author_sort | Zainidin Eshkuvatov |
| collection | DOAJ |
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This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integrodifferential equations (FracVF-IDEs). Other than that, the existance and uniqueness of solution as well as the norm convergence with respect to ND-HAM, were proven in a Hilbert space. In addition, three numerical examples (including multi-term fractional IDEs) are presented and compared with the HAM, modified HAM and ”Generalized block pulse operational differentiation matrices method” developed in previous works by illustrating the accuracy as well as validity with respect to ND-HAM. Empirical investigations reveal that ND-HAM and the modified HAM yields the same results when control parameter ℏ is chosen as ℏ = −1 and is comparable to the standard HAM. The findings discovered that the ND-HAM is highly convenient, effective, as well as in line with theoretical results.
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| format | Article |
| id | doaj-art-52316b8af5354853b2091c85c10e3854 |
| institution | Kabale University |
| issn | 1392-6292 1648-3510 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj-art-52316b8af5354853b2091c85c10e38542025-01-27T16:30:18ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102025-01-0130110.3846/mma.2025.20268Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAMZainidin Eshkuvatov0Zaid Laadjal1Shahrina Ismail2Faculty of Computer Science and Mathematics, University Malaysia Terengganu (UMT), Kuala Nerus, Terengganu, Malaysia; Faculty of Applied Mathematics and Intellectual Technology, National University of Uzbekistan (NUUz), Tashkent, UzbekistanInstitute of Sciences, University Center of Illizi, 33000 Illizi, AlgeriaFaculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), Negeri Sembilan, Malaysia This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integrodifferential equations (FracVF-IDEs). Other than that, the existance and uniqueness of solution as well as the norm convergence with respect to ND-HAM, were proven in a Hilbert space. In addition, three numerical examples (including multi-term fractional IDEs) are presented and compared with the HAM, modified HAM and ”Generalized block pulse operational differentiation matrices method” developed in previous works by illustrating the accuracy as well as validity with respect to ND-HAM. Empirical investigations reveal that ND-HAM and the modified HAM yields the same results when control parameter ℏ is chosen as ℏ = −1 and is comparable to the standard HAM. The findings discovered that the ND-HAM is highly convenient, effective, as well as in line with theoretical results. https://gc.vgtu.lt/index.php/MMA/article/view/20268homotopy analysis method (HAM)new development of HAM (ND-HAM)integro-differential equation (IDEs)Caputo fractional derivativeconvergence |
| spellingShingle | Zainidin Eshkuvatov Zaid Laadjal Shahrina Ismail Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM Mathematical Modelling and Analysis homotopy analysis method (HAM) new development of HAM (ND-HAM) integro-differential equation (IDEs) Caputo fractional derivative convergence |
| title | Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM |
| title_full | Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM |
| title_fullStr | Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM |
| title_full_unstemmed | Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM |
| title_short | Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM |
| title_sort | solving class of mixed nonlinear multi term fractional volterra fredholm integro differential equations by new development of ham |
| topic | homotopy analysis method (HAM) new development of HAM (ND-HAM) integro-differential equation (IDEs) Caputo fractional derivative convergence |
| url | https://gc.vgtu.lt/index.php/MMA/article/view/20268 |
| work_keys_str_mv | AT zainidineshkuvatov solvingclassofmixednonlinearmultitermfractionalvolterrafredholmintegrodifferentialequationsbynewdevelopmentofham AT zaidlaadjal solvingclassofmixednonlinearmultitermfractionalvolterrafredholmintegrodifferentialequationsbynewdevelopmentofham AT shahrinaismail solvingclassofmixednonlinearmultitermfractionalvolterrafredholmintegrodifferentialequationsbynewdevelopmentofham |