In this paper, two-dimensional Legendre wavelets algorithm is applied for the first time to solve a variable order fractional partial differential governing equation of viscoelastic column in the time domain. Firstly, the governing equation of a viscoelastic column is established according to a variable order fractional constitutive model. Secondly, the unknown function is expanded into the elements of two-dimensional Legendre wavelets. In order to obtain the numerical solutions of this type of equation, the differential operator matrices based on Legendre wavelets of integer order and variable order fractional are derived. The operator matrices are used to convert the initial governing equation into algebraic equations that are easy to solve in the time domain. The efficiency and accuracy of the algorithm are verified through the convergence analysis of Legendre wavelets and the error estimations of numerical example. Finally, the displacement solutions of the viscoelastic column under constant load and variable load are considered, and the columns with different cross-section shapes are studied. The results show that the proposed numerical algorithm is efficient in dynamic analysis of viscoelastic columns.

本文首次应用二维勒让德小波算法求解粘弹性柱的变阶分数阶偏微分控制方程。首先，根据变阶分数本构模型建立粘弹性柱的控制方程。其次，将未知函数展开为二维勒让德小波的元素。为了得到这类方程的数值解，推导了基于整数阶和变阶分数阶Legendre小波的微分算子矩阵。算子矩阵用于将初始控制方程转换为易于在时域中求解的代数方程。通过勒让德小波的收敛性分析和数值算例的误差估计，验证了算法的有效性和准确性。最后，考虑了粘弹性柱在恒载和变载下的位移解，研究了不同截面形状的柱。结果表明，所提出的数值算法在粘弹性柱的动力分析中是有效的。