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For the numerical solution of Dirichlet-type boundary value problems associated to nonlinear fractional differential equations of order \(\alpha \in (1,2)\) that use Caputo derivatives, we suggest to employ shooting methods. In particular, we demonstrate that the so-called proportional secting technique for selecting the required initial values leads to numerical schemes that converge to high accuracy in a very small number of shooting iterations, and we provide an explanation of the analytical background for this favourable numerical behaviour.

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Shooting Methods for Fractional Dirichlet-Type Boundary Value Problems of Order \(\alpha \in (1,2)\) with Caputo Derivatives

  • 会议论文

  • 发表日期：

    2025年

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  • Kai Diethelm

摘要

For the numerical solution of Dirichlet-type boundary value problems associated to nonlinear fractional differential equations of order \(\alpha \in (1,2)\) that use Caputo derivatives, we suggest to employ shooting methods. In particular, we demonstrate that the so-called proportional secting technique for selecting the required initial values leads to numerical schemes that converge to high accuracy in a very small number of shooting iterations, and we provide an explanation of the analytical background for this favourable numerical behaviour.