Nonlinear Dynamics

, Volume 38, Issue 1–4, pp 207–220

# Role of Prehistories in the Initial Value Problems of Fractional Viscoelastic Equations

• Masataka Fukunaga
Article

## Abstract

The fractional viscoelastic equation (FVE), which is a second-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 prehistories of the unknown functions before the initial times, referred to as the initial functions, are taken into account to solve FVEs. Mathematically, appropriate initial functions are essential for unique solutions of FVEs. Physically, the initial functions reflect the processes of giving the initial values. FVEs are solved for some initial functions both by analytical and numerical methods. The initial functions affect the solutions of FVEs. It is discussed how the solutions depend on the initial functions. Implication of the solutions to viscoelastic materials will be discussed.

## Key words:

initial condition initial function fractional differential equation fractional operators viscoelastic damping

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