多阶分数阶时滞微分方程的谱延迟校正法

Spectral delay correction method for multi-order fractional delay differential equations

分数阶时滞微分方程(FDDEs)在物理、生物等众多领域有着广泛应用。针对分数阶时滞微分方程(FDDEs),创造性地提出并应用谱延迟校正法(SDC)作为解决方案,构建一种基于双网格的Legendre延迟校正谱方法。引入双网格技术,对时间和空间离散进行优化处理,同时结合Legendre多项式进行谱延迟校正,大幅提升求解精度。制定预测步骤和校正步骤进行详尽误差分析。预测步骤以初步逼近的方式为解提供初始估计,通过校正步骤进一步细化解的近似,从而显著提高整体数值精度。数值实验结果表明,双网格Legendre延迟校正谱方法在处理分数阶时滞微分方程时成效卓著,极大地提高了精度,充分验证了理论结果的正确性。

Fractional delay differential equations(FDDEs)have extensive applications in diverse fields such as physics and biology.The spectral delay correction(SDC)method for FDDEs was proposed as a solution,and a double-grid-based Legendre delay correction spectral method was developed.The double-grid technique was introduced to optimize the time and space discretization,while the Legendre polynomial was used for spectral delay correction,significantly enhancing the solution accuracy.A detailed error analysis was conducted through prediction and correction steps.The prediction step provided an initial estimate for the solution through an approximation,while the correction step further refined the approximation,thereby substantially improving the overall numerical accuracy.Numerical experiments demonstrate that the double-grid Legendre delay correction spectral method performed exceptionally well in solving FDDEs,greatly improving accuracy and fully verifying the correctness of the theoretical results.