A multiscale Abel kernel and application in viscoelastic problem

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.

Figures (4)

• Figure 1.1: Log-log plots of (left) $k(t)$ in (\ref{['k']}) with $\alpha(t) = 0.9 + 0.1 e^{-0.1t}$ and its asymptotics in (\ref{['kzz1']}) and (right) $k(t;a)$ with $\alpha(t;a) = 0.7 + 0.3 e^{-at}$ for different $a$ and the Mittag-Leffler kernel $k_E(t)$.
• Figure 1.2: Plots of $\partial_t u(0.5,t)$ under different $\alpha(t)$.
• Figure 5.1: Solution curves of $u(0.5,t)$ under different parameters.
• Figure 5.2: Solution curves of $u(5,t)$ under different kernels.

Theorems & Definitions (12)

• Lemma 3.1
• Theorem 3.2
• Lemma 3.3
• Theorem 3.4
• Theorem 3.5
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof