A Matlab toolbox for positive fractional time derivative modeling of arbitrarily frequency-dependent viscosity

Abstract

This paper introduces a new Matlab toolbox for the numerical solution of power law frequency-dependent damped vibration and dissipative wave equations involving the positive fractional time derivative. The classical integer-order derivative models have long encountered huge difficulty in describing such complex thermoviscous behaviors, particularly if a broadband excitation is involved. Recent decades have seen a growing interest in the fractional derivative modeling of such anomalous viscosity. Among various time fractional derivative models, the positive fractional derivative model has clear advantages to hold causality of wave problems thanks to its positive definition. However, the numerical methods and software available today are mostly for the standard fractional derivative equations. This paper will present a finite difference method for positive fractional vibration and acoustic equations, and then focus on a new Matlab toolbox of its implementations, which is very easy to use with a friendly graphical user interface. The toolbox is freely available as an open source software and will help promote the application of the positive fractional derivative models to diverse dynamic problems where the viscosity plays an essential role.