A variable-order fractional differential equation model of shape memory polymers

Abstract

A shape-memory polymer (SMP) is capable of memorizing its original shape, and can acquire a temporary shape upon deformation and returns to its permanent shape in response to an external stimulus such as a temperature change. SMPs have been widely used industrial and medical applications. Previously, differential equation models were developed to describe SMPs and their applications. However, these models are often of very complicated form, which require accurate numerical simulations.In this paper we argue that a variable-order fractional differential equation model of the shape-memory behavior is more suitable than constant-order fractional differential equation models in terms of modeling the memory behavior of SMPs. We develop a numerical method to simulate the variable-order model and, in particular, to identify the unknown variable order of the model. Numerical experiments are presented to show the utility of the method.