Analytical examples of diffusive waves generated by a traveling wave

We construct analytical solutions for a system composed of a reaction–diffusion equation coupled with a purely diffusive equation. The question is to know if the traveling wave solutions of the reaction–diffusion equation can generate a traveling wave for the diffusion equation. Our motivation comes from the calcic wave, generated after fertilization within the egg cell endoplasmic reticulum, and propagating within the egg cell. We consider both the monostable (Fisher–KPP type) and bistable cases. We use a piecewise linear reaction term so as to build explicit solutions, which leads us to compute exponential tails whose exponents are roots of second-, third-, or fourth-order polynomials. These raise conditions on the coefficients for existence of a traveling wave of the diffusion equation. The question of positivity and monotonicity is only partially answered.     

Existence and Exponential Decay of Solutions for a Wave Equation with Integral Nonlocal Boundary Conditions of Memory Type

In this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, we establish two local existence theorems by using Faedo–Galerkin method and standard arguments of density. Next, we give a su?cient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

General decay of solutions of a wave equation with memory term and acoustic boundary condition

In this paper, the global solvability to the mixed problem involving the wave equation with memory term and acoustic boundary conditions for non‐locally reacting boundary is considered. Moreover, the general decay of the energy functionality is established by the techniques of Messaoudi. Copyright © 2016 John Wiley & Sons, Ltd.     

On the decay and blow‐up of solution for coupled nonlinear wave equations with nonlinear damping and source terms

In this work, we consider a nonlinear coupled wave equations with initial‐boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on the damping terms and source terms and initial data in the stable set, we obtain that the decay estimates of the energy function is exponential or polynomial by using Nakao's method. By using the energy method, we obtain the blow‐up result of solution with some positive or nonpositive initial energy. Copyright © 2016 John Wiley & Sons, Ltd.     

On mechanical waves and Doppler shifts from moving boundaries

We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, X_s(t)=vt is the location of the source with constant velocity v. In the present work, however, we consider a source co‐located with a moving boundary x=X_s(t), where X_s(t) can have an arbitrary functional form. For ‘slowly moving’ boundaries (i.e., ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple‐scale asymptotic analysis of the moving boundary problem for the linear wave equation. We obtain a closed‐form leading‐order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i) for non‐uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the non‐uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated. Copyright © 2017 John Wiley & Sons, Ltd.     

Exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions

By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Intrinsic decay rate estimates for abstract wave equation with memory

In this paper, we consider an abstract wave equation in the presence of memory. The viscoelastic kernel g(t) is subject to a general assumption , where the function H(·)∈C¹(R⁺) is positive, increasing and convex with H(0)=0. We give the decay result as a solution to a given nonlinear dissipative ODE governed by the function H(s). Copyright © 2017 John Wiley & Sons, Ltd.     

Asymptotic nonlinear stability of a composite wave of two traveling waves to a chemotaxis model

In this paper, we investigate the asymptotic stability of a composite wave consisting of two traveling waves to a Keller–Segel chemotaxis model with logarithmic sensitivity and nonzero chemical diffusion. We show that the composite wave is asymptotically stable under general initial perturbation, which only be needed small in H¹‐norm. This improves previous results. Copyright © 2017 John Wiley & Sons, Ltd.     

Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.