Analytical and Numerical Solutions for Fractional Viscoelastic Equations

Abstract

The fractional viscoelastic equation (FVE), which is a 2nd-order differential equation with fractional derivatives describing the dynamical behavior of a single-degree-of-freedom viscoelastic oscillator, is considered. Some viscoelastic damped mechanical systems may be described by FVEs. However, FVEs with conventional nonzero initial values cannot generally be solved. In this paper, the prehistory of the unknown before the initial time, referred to as the initial function, is taken into account in order to solve a FVE. Appropriate initial functions are essential for unique solutions of FVEs. Physically, the initial function reflects the process of giving the initial values. If a viscoelastic material is described by a FVE, the behavior of the material of the same initial values depends on the process of giving the initial values. FVEs are solved for some initial functions both by analytical and numerical methods. Implication of the solutions to viscoelastic materials will also be discussed.