ABSTRACT

In this paper, we study the spatial variable coefficient fractional convection–diffusion wave equation with the singe delay and multi-delay numerically when the exact solution satisfies a certain regularity. First, via the well-known exponential transformation, the delay problem can be greatly simplified, which allows us to use the variable coefficient four-order compact operator. Next, the numerical scheme is derived based on the compact operator and the reduction order method, followed by a linearized technique. Convergence of the full discrete numerical scheme is obtained with convergence order $O({\tau }^{3-\alpha }+h^4)$ under the maximum norm by the energy argument. We prove the almost unconditional stability of the scheme under very mild conditions. Extending the numerical method to the multi-delay case is available. Extensive computational results are presented including single delay and double delay problems, which demonstrate the effectiveness and correctness of the developed schemes.

摘要

本文对精确解满足一定规律时具有单时滞和多时滞的空间变系数分数对流-扩散波动方程进行数值研究。首先，通过众所周知的指数变换，可以大大简化延迟问题，这允许我们使用变系数四阶紧算子。接下来，基于紧算子和降阶方法推导出数值方案，然后是线性化技术。用收敛阶获得全离散数值格式的收敛$○({\tau }^{3-\alpha }+H^4)$在能量参数的最大范数下。我们证明了该方案在非常温和的条件下几乎无条件的稳定性。可以将数值方法扩展到多延迟情况。给出了广泛的计算结果，包括单延迟和双延迟问题，证明了所开发方案的有效性和正确性。