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<p>Fractional derivatives, due to their nonlocality, can describe complex processes where historical data is important for future calculations. At the same time, this property brings difficulties in numerical simulations. This paper presents a new discrete operator for approximating the fractional derivative based on the Grünwald–Letnikov definition, the “short memory principle,” memorization and analytical assumptions. This operator significantly reduces the number of operations in the calculation process when solving boundary value problems by saving the calculated data and transforming it for further use with adjustable accuracy. The results obtained showed that the use of this technique on specific problems reduced the execution time from ~ 1262&#xa0;s to ~ 19&#xa0;s, which is more than 66 times less than the standard method. It should be taken into account that the use of the modified method showed a worse result compared to the analytical method and amounted to ~ 0.011%.</p>

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Optimization algorithm for the numerical implementation of the fractional derivative of Grünwald–Letnikov based on the principle of memorization for fractional differential equations

  • 期刊论文

  • 发表日期：

    2025年10月25日

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  • Alibek Issakhov,
  • Aizhan Abylkassymova,
  • Stanislav Yun

摘要

Fractional derivatives, due to their nonlocality, can describe complex processes where historical data is important for future calculations. At the same time, this property brings difficulties in numerical simulations. This paper presents a new discrete operator for approximating the fractional derivative based on the Grünwald–Letnikov definition, the “short memory principle,” memorization and analytical assumptions. This operator significantly reduces the number of operations in the calculation process when solving boundary value problems by saving the calculated data and transforming it for further use with adjustable accuracy. The results obtained showed that the use of this technique on specific problems reduced the execution time from ~ 1262 s to ~ 19 s, which is more than 66 times less than the standard method. It should be taken into account that the use of the modified method showed a worse result compared to the analytical method and amounted to ~ 0.011%.