The diffraction of Kelvin wave due to an island

This paper describes a numerical method to determine the diffraction effect of an oceanographic Kelvin wave system by an island. A standard shallow water approximation theory is assumed for the oceanographic equation. The problem is solved in two cases, when the island is elliptic and circular in shape and when it is far or near to the coastline, also when the island becomes narrow and perpendicular to an infinitely long coastline. Numerical computation of diffraction of the Kelvin waves has been calculated and plotted.     

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

Approximate solutions for magmon propagation from a reservoir

^** Email: tim_marchant{at}uow.edu.au^*** Email: n.smyth{at}ed.ac.uk A 1D partial differential equation (pde) describing the flowof magma in the Earth's mantle is considered, this equationallowing for compaction and distension of the surrounding matrixdue to the magma. The equation has periodic travelling wavesolutions, one limit of which is a solitary wave, called a magmon.Modulation equations for the magma equation are derived andare found to be either hyperbolic or of mixed hyperbolic/elliptictype, depending on the specific values of the wave number, meanheight and amplitude of the underlying modulated wave. The periodicwave train is stable in the hyperbolic case and unstable inthe mixed case. Solutions of the modulation equations are foundfor an initial-boundary value problem on the semi-infinite line,these solutions representing the influx of magma from a largereservoir. The modulation solutions are found to consist ofa full or partial undular bore. Excellent agreement with numericalsolutions of the governing pde is obtained, except in the limitwhere the wave train becomes a train of magmons. An alternativeapproximation based on the assumption that the wave train isa series of uniform magmons is also derived and is found tobe superior to modulation theory in this limit.     

The inverse electromagnetic scattering problem for a partially coated dielectric

We use the linear sampling method to determine the shape and surface conductivity of a partially coated dielectric infinite cylinder from knowledge of the far field pattern of the scattered TM polarized electromagnetic wave at fixed frequency. A mathematical justification of the method is provided based on the use of a complete family of solutions. Numerical examples are given showing the efficiency of our method.     

Hilbertian spatial periodically correlated first order autoregressive models

In this article, we consider Hilbertian spatial periodically correlated autoregressive models. Such a spatial model assumes periodicity in its autocorrelation function. Plausibly, it explains spatial functional data resulted from phenomena with periodic structures, as geological, atmospheric, meteorological and oceanographic data. Our studies on these models include model building, existence, time domain moving average representation, least square parameter estimation and prediction based on the autoregressive structured past data. We also fit a model of this type to a real data of invisible infrared satellite images.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the problem of decay rate for the solutions of the initial-boundary value problem to the wave equation, governed by localized nonlinear dissipation and without any assumption on the dynamics (i.e., the control geometric condition is not satisfied). We treat separately the autonomous and the non-autonomous cases. Providing regular initial data, without any assumption on an observation subdomain, we prove that the energy decays at last, as fast as the logarithm of time. Our result is a generalization of Lebeau (in: A. Boutet de Monvel, V. Marchenko (Eds.), Algebraic and Geometric Methods in Mathematical Physics, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1996, pp. 73) result in the autonomous case and Nakao (Adv. Math. Sci. Appl. 7 (1) (1997) 317) work in the non-autonomous case. In order to prove that result we use a new method based on the Fourier-Bross-Iaglintzer (FBI) transform.     

On transmission problem for viscoelastic wave equation with a localized a nonlinear dissipation

In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials, one component being a Kirchhoff type wave equation with time dependent localized dissipation which is effective only on a neighborhood of certain part of boundary, while the other being a Kirchhoff type viscoelastic wave equation with nonlinear memory.     

Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I

The shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations [A.M. Blokhin, On stability of shock waves in a compressible viscous gas, Matematiche LVII (I) (2002) 3-19]. We suggest such additional boundary conditions that a modified (with account to these conditions) linear initial-boundary value problem on stability of the shock wave does not admit Hadamard-type ill-posedness examples.     

ASYMPTOTIC BEHAVIOR TOWARD THERAREFACTION WAVE FOR SOLUTIONS OFRATE-TYPE VISCOELASTIC SYSTEM WITH BOUNDARY EFFECT

1IntroductionWeconsiderthefollowinginitialboundaryvalueproblemonR =(o, oo)forarate-typeviscoelasticsystemwiththeinitial-boundaryconditionsWherev5uand-pdenotethestrain,partialvelocityandstressrespectively,whileEisapositiveconstant,whichrepresentthedynamicYoung'smodulus,andT>Oisarelaxationtime.Forsimplicity,weassumeT=1.PR(v)standsfortheequilibriumvalueforp-Theinitialdata(vo,bolpo)(x)areassumedtotendtotheconstantstate,asx- oowherep =pR(v )sincepR(v)istheequilibriumvaluef0rp.M0reover,thecompa…     

The configuration of shock wave reflection for the TSD equation

In this paper, we mainly study the nonlinear wave configuration caused by shock wave reflection for the TSD (Transonic Small Disturbance) equation and specify the existence and nonexistence of various nonlinear wave configurations. We give a condition under which the solution of the RR (Regular reflection) for the TSD equation exists. We also prove that there exists no wave configuration of shock wave reflection for the TSD equation which consists of three or four shock waves. In phase space, we prove that the TSD equation has an IR (Irregular reflection) configuration containing a centered simple wave. Furthermore, we also prove the stability of RR configuration and the wave configuration containing a centered simple wave by solving a free boundary value problem of the TSD equation.     

ON TRANSMISSION PROBLEM FOR KIRCHHOFF TYPE WAVE EQUATION WITH A LOCALIZED NONLINEAR DISSIPATION IN BOUNDED DOMAIN＊

In this article,we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials,one...     

Existence and stability of traveling wave solutions to one‐sided mixed initial‐boundary value problem for first‐order quasilinear hyperbolic systems

When one characteristic of the system is linearly degenerate, under suitable boundary conditions, we get the existence of traveling wave solutions located on the corresponding characteristic trajectory to the one‐sided mixed initial‐boundary value problem. When the system is linearly degenerate, by introducing the semi‐global normalized coordinates, we derive the related formulas of wave decomposition to prove the stability of traveling wave solutions corresponding to all leftward and the rightmost characteristic trajectories. Finally, for the traveling wave solutions corresponding to other rightward characteristic trajectories, some examples show their possible instability. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical computation of sharp travelling waves of a degenerate diffusion–reaction equation arising in biofilm modelling

Many degenerate diffusion–reaction equations permit sharp travelling wave solutions that describe the propagation of an interface with finite speed. If the equation is at least double degenerate, the derivative of the travelling wave solution can blow up at the interface, which poses considerable challenges for the computation of the travelling wave speed. We propose a numerical method for this problem that is based on the idea to approximate the multiple degenerate problem by a family of simple degenerate problems. For the latter we propose an interval-bracketing algorithm based on the theory of Sanchez-Garduno and Maini. The travelling wave speed of the original problem is obtained as the limit of the travelling wave speeds of the auxiliary problems. The performance of the method is investigated in a numerical simulation experiment for a problem that arises in the mathematical modelling of biofilm processes.     

Elastic waves in regions with multiple interfaces

The purpose of this article is to illustrate how the study of partial differential equations (PDEs) can be made more accessible to undergraduates through the use of applications and computers. The analysis initially concentrates on travelling waves on strings and their reflections from boundaries. Then wave behaviour on strings with multiple interfaces, where incident waves are repeatedly reflected and transmitted, is investigated. Finally, the results are extended to waves in three-dimensional bodies in which longitudinal and transverse waves occur simultaneously. The effectiveness of a computer algebra system (CAS) for visualizing and understanding solutions is demonstrated. In particular, snapshots and sample code for Mathematica animations are provided to demonstrate the physical nature of the solutions. Although intended for PDE students, the contents of this paper are accessible to undergraduates who have studied ordinary differential equations (ODEs) and would be appealing to anyone with a natural curiousity about waves.     

Stability of the planar rarefaction wave to two-dimensional Navier-Stokes-Korteweg equations of compressible fluids

This study is concerned with the large time behavior of the two-dimensional compressible Navier-Stokes-Korteweg equations, which are used to model compressible fluids with internal capillarity. Based on the fact that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws is nonlinearly stable to the one-dimensional compressible Navier-Stokes-Korteweg equations, the planar rarefaction wave to the two-dimensional compressible Navier-Stokes-Korteweg equations is first derived. Then, it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are suitably small perturbations of the planar rarefaction wave. The proof is based on the delicate energy method. This is the first stability result of the planar rarefaction wave to the multi-dimensional viscous fluids with internal capillarity.     

Exponential Decay of Energy for a Logarithmic Wave Equation

In this paper we consider the initial boundary value problem for a class of logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for the logarithmic wave equation with linear damping and some suitable initial data. The results extend the early results.     

Exact boundary observability for 1‐D quasilinear wave equations

By means of a direct and constructive method based on the theory of semi‐global C² solution, the local exact boundary observability and an implicit duality between the exact boundary controllability and the exact boundary observability are shown for 1‐D quasilinear wave equations with various boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

对波动方程初边值问题边界条件齐次化函数的一个注解

基于传统的齐次化边界条件方法,采用傅里叶级数法讨论了波动方程初边值问题第一类非齐次边界条件齐次化函数问题,分析表明:对同一定解问题,在不同齐次化函数下的解在适定意义下是等价的.