Seismic wave propagation in viscoelastic media under Atangana-Baleanu fractional dynamics: Model formulation and numerical simulations

TL;DR

The paper develops a 1D fractional viscoelastic wave model for seismic propagation using the Atangana–Baleanu–Caputo derivative with a non-singular Mittag–Leffler kernel to capture memory effects. It couples a constitutive law with an ABC derivative to yield a fractional wave equation, discretizes in space with centered finite differences, and advances in time via an Adams–Bashforth–Moulton predictor–corrector scheme. The main findings show that memory parameters $\alpha$ and $\kappa$ modulate attenuation and dispersion, producing non-exponential (Mittag–Leffler) energy decay and smoother wave-packet dispersion compared to the classical case. The work demonstrates a robust numerical framework implemented in Python suitable for extension to heterogeneous media and data-driven parameter estimation, offering a tractable path toward more realistic seismic attenuation models.

Abstract

We propose a one-dimensional viscoelastic seismic-wave model driven by the Atangana-BaleanuCaputo fractional derivative with a non-singular Mittag-Leffler kernel. A finite-difference discretization in space and an Adams-Bashforth-Moulton predictor-corrector scheme in time are used to compute solutions for several fractional orders. Simulations indicate that fractional memory alters both attenuation and dispersion, leading to non-exponential energy decay compared with the classical integer-order case.

Figures (4)

• Figure 1: The modal amplitude is time dependent, where $A(t)=E_{\alpha}(-\lambda t^{\alpha})$ shows different behaviours with respect to the order of fractional decay $\alpha$. The case of $\alpha=1$ shows only an exponential decay. Both cases for $\alpha=0.8$ and $\alpha=0.6$ exhibit Mittag-Leffler type decay (or relaxation) but at a much slower rate (i.e., more heavily tailed), indicating that a fractional memory effect is contributing significantly to the behaviour of the decay.
• Figure 2: Semi-logarithmic plots of modal energy $E(t)=\tfrac{1}{2}(A'(t)^2+\pi^2 A(t)^2)$ The energy behaves like a linear function when plotted with respect to the logarithm of time because it decays almost exactly in an exponential manner when $\alpha=1$. On the other hand, both curves for $\alpha=0.8$ and for $\alpha=0.6$ decay in a slower, non-exponential manner (Mittag-Leffler) due to the enhanced memory effects of the fractional order kernel.
• Figure 3: The Space-time contour plot shows the way the displacement $w(x,t)=A(t)\sin(\pi x)$ behaves over the course of time for a representative fractional order (here $\alpha=0.8$). The fact that the endpoints are held fixed (i.e. tested using the condition $w(0,t)=w(1,t)=0$) is evident. The main feature of the interior is the formation and decay of a standing wave as the amplitude decays over time through the Mittag-Leffler type of the decay function generated by the Atangana-Baleanu fractional kernel.
• Figure 4: Displacement, $w(x,t)$, shown in a 3D graph with values defined by the equation of a standing wave based on the fixed support conditions; Also shown is the amplitude of the wave progressively decreasing over time based on the fractional order differential equation created through the Mittag-Leffler type relaxation introduced by the Atangana-Baleanu fractional derivative.