Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear
In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key param...
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| Format: | Article |
| Language: | English |
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AIMS Press
2021-02-01
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| Series: | AIMS Mathematics |
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| Online Access: | http://www.aimspress.com/article/doi/10.3934/math.2021225?viewType=HTML |
| _version_ | 1818330051438968832 |
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| author | Khaled M. Saad Manal Alqhtani |
| author_facet | Khaled M. Saad Manal Alqhtani |
| author_sort | Khaled M. Saad |
| collection | DOAJ |
| description | In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, 1<q<2, in the equation on the behavior of numerical solutions of fractal-fractional reaction diffusion equations (FFRDE). Iterative approximations to the solutions of these equations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In key parameter regimes, all these iterative solutions based on a power kernel, an exponential kernel and a generalized Mittag-Leffler kernel are very close. Hence, iterative solutions obtained using one of these kernels are compared with full numerical solutions of the FFRDE and excellent agreement is found. All numerical solutions in this paper were obtained using Mathematica. |
| first_indexed | 2024-12-13T12:57:48Z |
| format | Article |
| id | doaj.art-32966b98724b4fb19c914badc596e5f4 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-13T12:57:48Z |
| publishDate | 2021-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-32966b98724b4fb19c914badc596e5f42022-12-21T23:45:08ZengAIMS PressAIMS Mathematics2473-69882021-02-01643788380410.3934/math.2021225Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinearKhaled M. Saad0Manal Alqhtani11. Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Kingdom of Saudi Arabia 2. Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen1. Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Kingdom of Saudi ArabiaIn this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, 1<q<2, in the equation on the behavior of numerical solutions of fractal-fractional reaction diffusion equations (FFRDE). Iterative approximations to the solutions of these equations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In key parameter regimes, all these iterative solutions based on a power kernel, an exponential kernel and a generalized Mittag-Leffler kernel are very close. Hence, iterative solutions obtained using one of these kernels are compared with full numerical solutions of the FFRDE and excellent agreement is found. All numerical solutions in this paper were obtained using Mathematica.http://www.aimspress.com/article/doi/10.3934/math.2021225?viewType=HTMLthe fractal-fractional reaction diffusion equationslagrange polynomial interpolationthe power lawthe exponential lawgeneralized mittag-leffler function |
| spellingShingle | Khaled M. Saad Manal Alqhtani Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear AIMS Mathematics the fractal-fractional reaction diffusion equations lagrange polynomial interpolation the power law the exponential law generalized mittag-leffler function |
| title | Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear |
| title_full | Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear |
| title_fullStr | Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear |
| title_full_unstemmed | Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear |
| title_short | Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear |
| title_sort | numerical simulation of the fractal fractional reaction diffusion equations with general nonlinear |
| topic | the fractal-fractional reaction diffusion equations lagrange polynomial interpolation the power law the exponential law generalized mittag-leffler function |
| url | http://www.aimspress.com/article/doi/10.3934/math.2021225?viewType=HTML |
| work_keys_str_mv | AT khaledmsaad numericalsimulationofthefractalfractionalreactiondiffusionequationswithgeneralnonlinear AT manalalqhtani numericalsimulationofthefractalfractionalreactiondiffusionequationswithgeneralnonlinear |