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The Time Evolution of the Random Genetic Drift Equation and the Computational Proof of Galton’s Theory |
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PP: 709-726 |
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doi:10.18576/pfda/090414
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Author(s) |
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Entsar Ahmed Abdel-Rehim,
Radwa Mohamed Hassan,
Ahmed Mohamed Ahmed El-Sayed,
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Abstract |
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| This paper focuses in studying the genetic diffusion process of a finite population in the absence of mutation, migration and immigration. In this paper, we aim to numerically solve the modified Kimura model in order to mathematically and computationally prove the Galton’s theory of hereditary genius. There are two steps to the proof. Assuming that the number of diploid individuals varies randomly over generations is the first step. In the second step, the classical random genetic drift equation with selection is extended to the time-fractional random genetic drift equation to include the memory effect on the diffusion process. The fourth order compact finite difference is implemented to derive the approximate solutions of the studied partial differential equation. For the purpose of determining the discrete scheme of the time-fractional operator, the Gru ̈nwald-Letnikov scheme is implemented. The simulations of the approximate solutions, the stationary approximate solution, the stability of the difference scheme and the total sum of the approximate solutions are all calculated and interpreted computationally. The discrete convergence of the approximate solutions is numerically studied and its asymptotic behaviors are compared with the Mittag-Leffler function.
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