In this study, torsional wave propagation is investigated in a rod that are made of one and two dimensional functionally graded material. Firstly, the governing equations of the wave propagation in the functionally graded cylinder derived in polar coordinate. Secondly, finite difference method is used to discretize the equations. The Von Neumann stability approach is used to obtain the time ste...

We extend the technique of optimal time step (OTS) selection for finite difference (FD) schemes of time dependent PDEs to PDEs where the leading-order spatial derivative term has a spatially varying coefficient. The basic approach involves identifying a transformation of the domain that eliminates the spatial dependence of the coefficient for the leading-order term. This change of variables is ...

In this paper, we study the variable-order (VO) time-fractional diffusion equations. For a VO function $$\alpha (t)\in (0,1)$$ , develop an exponential-sum-approximation (ESA) technique to approach Caputo fractional derivative. The ESA keeps both quadrature exponents and number of exponentials in summation unchanged at different time level. Approximating parameters are properly selected achieve...

As a mathematical tool, variable-order (VO) fractional calculus (FC) was developed rapidly in the engineering field due to it better describing anomalous diffusion problems engineering; thus, research of solutions VO differential equations (FDEs) has become hot topic for FC community. In this paper, we propose an effective numerical method, named as ε-approximate approach, based on least square...