We develop an indirect collocation method for a variable-order fractional wave equation. Fractional differential equations are well known to exhibit initial weak singularity, which makes it unrealistic to carry out error estimates of their numerical approximations based on the smoothness assumptions of the true solutions. In this paper, we analyze the convergence behaviour of the method without artificially assuming the (often untrue) full regularity of the true solution of the problem, but only based on the behaviour of the coefficients and variable order of the problem. More precisely, we prove the following results: (i) If the variable order has an integer initial value, the method discretized on a uniform partition has an optimal-order convergence rate in the $L_{\mathrm{\infty }}$ norm. (ii) Otherwise, the method discretized on a uniform mesh has only a sub-optimal order convergence rate. The method discretized on a graded mesh with the mesh grading parameter determined by the initial value of the variable order has an optimal-order convergence rate in the $L_{\mathrm{\infty }}$ norm. Numerical experiments are performed to substantiate the theoretical results.

我们为可变阶分数波动方程开发了一种间接搭配方法。众所周知，分数阶微分方程表现出初始弱奇异性，这使得基于真解的平滑性假设对其数值近似进行误差估计是不现实的。在本文中，我们分析了该方法的收敛行为，没有人为地假设问题的真实解的（通常是不真实的）完全正则性，而仅基于问题的系数和变量阶数的行为。更准确地说，我们证明了以下结果： (i) 如果变量阶数具有整数初始值，则在均匀分区上离散化的方法在${升}_{\mathrm{\infty }}$规范。(ii) 否则，在均匀网格上离散化的方法只有次优阶收敛速度。该方法在分级网格上离散化，网格分级参数由可变阶次的初始值确定，在${升}_{\mathrm{\infty }}$规范。进行数值实验以证实理论结果。