A multiscale Abel kernel and application in viscoelastic problem

TL;DR

This paper introduces a multiscale Abel kernel for modeling crossover dynamics in viscoelastic materials and develops a second-order numerical scheme with theoretical error analysis, validated by numerical experiments.

Contribution

It presents a novel multiscale variable-exponent Abel kernel and a second-order numerical scheme for viscoelastic problems with theoretical error estimates.

Findings

The kernel captures crossover from exponential to power-law behavior.

The numerical scheme achieves second-order accuracy.

Numerical results confirm theoretical predictions and model dynamics.

Abstract

We consider the variable-exponent Abel kernel and demonstrate its multiscale nature in modeling crossover dynamics from the initial quasi-exponential behavior to long-term power-law behavior. Then we apply this to an integro-differential equation modeling, e.g. mechanical vibration of viscoelastic materials with changing material properties. We apply the Crank-Nicolson method and the linear interpolation quadrature to design a temporal second-order scheme, and develop a framework of exponentially weighted energy argument in error estimate to account for the non-positivity and non-monotonicity of the multiscale kernel. Numerical experiments are carried out to substantiate the theoretical findings and the crossover dynamics of the model.