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<p>Numerical analysis of nabla <i>h</i>–fractional difference equations is provided in this paper. New numerical schemes are proposed using discrete fractional calculus. Furthermore, the relationship between fractional differential and difference equations is analyzed, and the convergence order is given. The asymptotic behavior of the nabla discrete Mittag-Leffler function is discussed, and the stable conditions for the numerical scheme of the time-fractional diffusion equation are explicitly given. It can be concluded that the discrete fractional calculus theory provides a non-standard discretization of fractional operators. Since there are no smooth assumptions or regularization conditions, the method can be extensively employed in initial value problems of fractional differential equations.</p>

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A time scale numerical method for fractional differential equations without initial value’s smooth assumptions

  • 期刊论文

  • 发表日期：

    2025年6月4日

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  • Hua Kong,
  • Guo-Cheng Wu,
  • Minfu Feng

摘要

Numerical analysis of nabla h–fractional difference equations is provided in this paper. New numerical schemes are proposed using discrete fractional calculus. Furthermore, the relationship between fractional differential and difference equations is analyzed, and the convergence order is given. The asymptotic behavior of the nabla discrete Mittag-Leffler function is discussed, and the stable conditions for the numerical scheme of the time-fractional diffusion equation are explicitly given. It can be concluded that the discrete fractional calculus theory provides a non-standard discretization of fractional operators. Since there are no smooth assumptions or regularization conditions, the method can be extensively employed in initial value problems of fractional differential equations.