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Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods

机译：谱配点法求解具有多个时滞的多阶分数阶微分方程的数值解

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Highlights•A framework for solving multi-delay fractional differential equations is proposed.•Fractional delay differential equations with irrational delays are discretized.•The method possesses spectral convergence with efficient computational time.•The convergence, error estimates, and numerical stability of the method are studied.•Several illustrative practical examples show the advantages of the method.AbstractThis paper discusses a general framework for the numerical solution of multi-order fractional delay differential equations (FDDEs) in noncanonical forms with irrational/rational multiple delays by the use of a spectral collocation method. In contrast to the current numerical methods for solving fractional differential equations, the proposed framework can solve multi-order FDDEs in a noncanonical form with incommensurate orders. The framework can also solve multi-order FDDEs with irrational multiple delays. Next, the framework is enhanced by the fractional Chebyshev collocation method in which a Chebyshev operation matrix is constructed for the fractional differentiation. Spectral convergence and small computational time are two other advantages of the proposed framework enhanced by the fractional Chebyshev collocation method. In addition, the convergence, error estimates, and numerical stability of the proposed framework for solving FDDEs are studied. The advantages and computational implications of the proposed framework are discussed and verified in several numerical examples.
机译： 突出显示 • 提出了求解多延迟分数阶微分方程的框架。 • 具有不合理延迟的分数延迟微分方程被离散化。 • 该方法具有频谱收敛性 • 研究了该方法的收敛性，误差估计和数值稳定性。 • 几个说明性的实际示例展示了该方法的优点。 摘要 本文讨论了多数值解的通用框架非规范形式的无序/有理多重延迟的三阶分数阶延迟微分方程（FDDE），这是通过使用频谱搭配方法来实现的。与当前求解分数阶微分方程的数值方法相比，所提出的框架可以求解非规范形式且阶数不等的多阶FDDE。该框架还可以解决具有非理性多重延迟的多阶FDDE。接下来，通过分数切比雪夫搭配方法增强框架，在该方法中，构造了切比雪夫运算矩阵用于分数微分。谱收敛和计算时间短是分数Chebyshev配置方法增强了所提出框架的另外两个优点。此外，研究了所提出的FDDEs框架的收敛性，误差估计和数值稳定性。在几个数值示例中讨论并验证了所提出框架的优点和计算含义。

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