Initial Condition Problems of Fractional Viscoelastic Equations

Abstract

The 2nd order differential equation with fractional derivatives describing dynamic behavior of a single-degree-of-freedom viscoelastic oscillator, referred to as fractional viscoelastic equation (FVE), is considered. Some types of viscoelastic damped mechanical systems may be described by FVE. The differential equation with fractional derivatives is often called the fractional differential equation (FDE). FDE can be solved for zero initial values, but it can not generally be solved for non-zero initial values. How to solve the problem is one of the key issues in this field. This is called “Initial condition (value) problems” of FDE. In this paper, initial condition problems of FVE are solved by making use of the prehistory functions of unknowns which are specified before the initial instance (referred to as the initial functions) starts. Introduction of initial functions into FDE reflects the physical state in giving the initial values. In this paper, several types of initial function are used to solve unique solutions for a type of FVE (referred to as FVE-I). The solutions of FVE-I are obtained by means of both numerical and analytical methods. Implication of the solutions to viscoelastic material will also be discussed.