Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott
A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transfor...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/2544688 |
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| author | Sami Aljhani Mohd Salmi Md Noorani Khaled M. Saad A. K. Alomari |
| author_facet | Sami Aljhani Mohd Salmi Md Noorani Khaled M. Saad A. K. Alomari |
| author_sort | Sami Aljhani |
| collection | DOAJ |
| description | A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ℏu,v-curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution. |
| format | Article |
| id | doaj-art-15c2e4c973894e74b7455ad61cbca74a |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-15c2e4c973894e74b7455ad61cbca74a2025-02-03T01:27:21ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/25446882544688Numerical Solutions of Certain New Models of the Time-Fractional Gray-ScottSami Aljhani0Mohd Salmi Md Noorani1Khaled M. Saad2A. K. Alomari3Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi Selangor, MalaysiaDepartment of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi Selangor, MalaysiaDepartment of Mathematics, College of Sciences and Arts, Najran University, POB 1988, Najran 11001, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Yarmouk University, 211-63 Irbid, JordanA reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ℏu,v-curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.http://dx.doi.org/10.1155/2021/2544688 |
| spellingShingle | Sami Aljhani Mohd Salmi Md Noorani Khaled M. Saad A. K. Alomari Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott Journal of Function Spaces |
| title | Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott |
| title_full | Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott |
| title_fullStr | Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott |
| title_full_unstemmed | Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott |
| title_short | Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott |
| title_sort | numerical solutions of certain new models of the time fractional gray scott |
| url | http://dx.doi.org/10.1155/2021/2544688 |
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