A rigorous derivation and energetics of a wave equation with fractional damping

Abstract

We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water–air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy–dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally damped wave equation with a time derivative of order 3/2.

Pseudo‐Lyapunov methods for Grünwald‐Letnikov and initialized fractional systems

“…To establish a Lyapunov method for system (1) in its initialized version, a functional $$$ E $$$ must be considered, which needs to include all the past values of $$$ x $$$ and hence, the function $$$ g $$$ of Assumption 1 should be known. The natural candidate for $$$ E $$$ is the integral of $$$ {x}^{\prime }x $$$, but once its first derivative is obtained, using system equations to get to the final stability conclusions is not straightforward.Instead, if we consider the functional $$$ E $$$ defined as $$$$ E(t)=\int_{r={t}_s}^t\int_{\tau ={t}_s}^t\frac{\alpha \Gamma {\left(1-\alpha \right)}^{-1}}{{\left(\tau +r\right)}^{\alpha }}{x}^{\prime }(r)x\left(\tau \right) d\tau dr, $$$$ it follows that $$$ E\ge 0 $$$ when $$$ x $$$ is continuous (see final part of Mielke et al 25 , Section 4.2 ) and, more important, the equations of system (1) can be directly used due to the following property …”

“…it follows that E ≥ 0 when x is continuous (see final part of Mielke et al 25,Section 4.2 ) and, more important, the equations of system (1) can be directly used due to the following property…”

“…Notice that V(t s ) = 0 and V(t) ≥ 0 for all t > t s . Thus, integrating the last inequality in (25), we obtain for any t > t 0…”

“…x ′ (𝜏)g(x(𝜏), 𝜏)d𝜏 < ∞ follows from Assumption 1, and this proves the first claim. On the other hand, integrating the second line in (25), we obtain…”