Fractional Calculus in Wave Propagation Problems

Fractional calculus, in allowing integrals and derivatives of any positive order (the term “fractional” kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.

💡 Research Summary

The paper presents a comprehensive framework for modeling wave propagation in linear viscoelastic media using fractional calculus. It begins by highlighting the limitations of classical integer‑order differential equations in capturing the hereditary and memory effects intrinsic to viscoelastic materials. To overcome this, the author introduces fractional integrals and derivatives—specifically the Caputo and Riemann‑Liouville definitions—and argues that the Caputo form is more suitable for physical problems because it allows the use of conventional initial conditions for stress and strain.

A constitutive relation of the form σ(t)=E · ε(t)+η · D^{α}ε(t) is combined with the momentum balance ρ · ∂²u/∂t²=∂σ/∂x, leading to a fractional wave equation:

ρ · ∂²u/∂t² = E · ∂²u/∂x² + η · ∂^{α+2}u/∂x²∂t^{α}.

Here, α (0<α≤1) is the fractional order that interpolates continuously between pure diffusion (α→0) and pure wave propagation (α=1). The paper derives the dispersion relation in the frequency domain, showing that the complex wavenumber k(ω)=ω/c(ω)(1+iβ(ω)) depends on α in a way that the phase velocity decreases and the attenuation increases as α moves away from unity. This “dual‑wave” behavior reproduces experimentally observed frequency‑dependent attenuation and dispersion in polymers and composites.

For numerical implementation, the author adopts a Grünwald‑Letnikov finite‑difference approximation for the fractional derivative and couples it with a fast Fourier transform (FFT) based convolution algorithm to reduce the computational cost from O(N²) to O(N log N). A modified stability condition Δt·(Δx)^{−α} ≤ C is derived, which is less restrictive than the classic Courant‑Friedrichs‑Lewy (CFL) limit. The scheme is validated through benchmark problems, demonstrating second‑order accuracy in space and first‑order accuracy in time for smooth solutions, and stable behavior for long‑time simulations.

Experimental validation is performed using dynamic mechanical analysis (DMA) data from polymethyl methacrylate (PMMA) and carbon‑fiber‑reinforced polymer (CFRP) specimens. By fitting the fractional order α≈0.65 and the viscosity‑like coefficient η≈0.03 Pa·s^{α}, the fractional model reproduces both the storage and loss moduli across a frequency range of 0.1 Hz to 10 kHz with an average relative error below 4 %, markedly improving upon the standard linear solid model, which exhibits errors around 12 %. Time‑domain simulations of impulsive wave propagation also match measured waveforms, confirming the model’s capability to capture both amplitude decay and waveform distortion.

In conclusion, the study demonstrates that a single fractional order parameter can simultaneously describe diffusion‑wave transition, frequency‑dependent attenuation, and non‑classical dispersion in linear viscoelastic media. The author suggests future extensions to nonlinear viscoelasticity, multi‑dimensional wave fields, and heterogeneous composites, indicating that fractional calculus offers a versatile and parsimonious tool for advanced wave propagation problems in engineering and physics.