Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results

We use a finite-difference scheme to simulate 2D ground penetrating radar data by solving the damped wave equation. We show that the stability condition for the damped scalar wave equation does not depend on the damping factor. As an example application of our modeling tool we investigate a typical situation in granite prospecting, where a conductive clay overburden masks the real position of the fractures in the granite. The results of the modeling have a good agreement with the actual data. Also, the algorithm shows well the increase in attenuation with the increase of the frequency. The algorithm shows to be a good basic modeling tool that can be used for further applications and for comparisons with other modeling methods like pseudospectral or finite elements. Introduction 2D forward numerical modeling of GPR data is an important tool to validate the interpretation of actual radargrams. Goodman (1994) and Cai and McMechan (1995) presented forward modeling routines using raytracing techniques. The drawback of these techniques is that they do not simulate diffraction events, as outlined by Zeng et al. (1995), who presented modeling by Fourier methods and compared them with the ray methods. Casper and Kung (1996) applied the pseudospectral forward modeling algorithm on GPR based on an explicit solution of the 2-D lossy electromagnetic wave equation and Chen and Huang (1996) used finite-differences following the same approach. All these methods have in common the assumption of a dielectric behavior of the earth, which is valid for most geologic applications of GPR, figuring in the frequency range between 10 and 1000 MHz, and imaging materials with conductivities lower than 100 mS/m (Davis and Annan, 1989). Under such conditions the velocity of propagation remains constant and the attenuation may be considered seperately from the velocity, what gives the electromagnetic radar waves the same behavior as the seismic acoustic waves. In the present work we use this electromagnetic-acoustic analogy and employ classical finite-differences to solve the damped scalar wave equation following Chen and Huang (1996) and Alford et. al. (1974). We show that the stability condition for the damped scalar wave equation does not depend on the damping factor, therefore reducing to the same condition presented by Alford et. al. (1974) for the wave equation without attenuation. Also, we study the effect of attenuation on radargrams by computing 2D synthetic zero-offset sections. As an example application of our modeling tool we investigate a typical situation in granite prospecting, where a conductive clay overburden masks the real position of the fractures in the granite. The results of the modeling have a good agreement with the actual data and show the usefulness of our approach. The algorithm shows to be a good basic modeling tool that can be used for further applications in migration or for comparisons with other modeling methods like the pseudospectral or finite elements. The Electromagnetic Wave Equation Our derivation of the 2D electromagnetic damped waveequation follows Casper and Kung (1996). Maxwell’s equations are t ∂ ∇× = − − ∂ s B E M (1) t ∂ ∇× = + + ∂ c D H J s J (2)

This work is devoted to the analysis of a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2-D domain. The damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero as time goes to infinity. Using discrete multiplier techniques, we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy for the solutions of the numerical scheme. The numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept, which guarantees that the low frequencies are damped correctly. Numerical experiments are presented and confirm these theoretical results. These results extend those by Tcheugoue-Tebou and Zuazua [ Numer. Math. 95 , 563–598 (2003)] where the 1-D case was addressed as well the square domain in 2-D. The methods and results in this paper extend to smooth domains in any space dimension.

In this paper we study well-posedness of the damped nonlinear wave equation in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ℝ2; ω⩾0, ωλ1+µ>0 with λ1 being the first eigenvalue of −Δ under zero boundary condition. Under the assumptions that g(·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of various classes of damped nonlinear wave equations. Specifically, stabilization the zero steady state solution of initial boundary value problems for nonlinear weakly and strongly damped wave equations, nonlinear wave equation with nonlinear damping term and some related nonlinear wave equations, introducing a feedback control terms that employ parameters, such as, finitely many Fourier modes, finitely many volume elements and finitely many nodal observables and controllers. In addition, we also establish the stabilization of the zero steady state solution to initial boundary value problem for the damped nonlinear wave equation with a controller acting in a proper subdomain. Notably, the feedback controllers proposed here can be equally applied for stabilizing other solutions of the underlying equations.

The main purpose of this article is to establish the Runge-type approximation in L^2(0,T;widetilde{H}^s(Omega )) for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calderón problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calderón problem of nonlocal wave equations with polyhomogeneous nonlinearities.

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

In this paper a general heat conduction law has been proposed based on the thermomass theory, which can be derived from the Boltzmann equation for phonons in dielectrics. The Boltzmann equation for phonons gives a balance between the drift and friction parts of the distribution function. When the normal scattering term is omitted in the friction part, the Fourier’s law and Cattaneo-Vernotte thermal wave equation can be obtained by the zeroth order and first order approximations of the drift term, respectively. A second order approximation of the drift part lead to the thermomass theory based general heat conduction law, which is a nonlocal damped heat wave equation and consists of driving, inertial and resistant forces for phonon gas motion. In nanosystems, the normal scattering term of the friction part reflecting the boundary effect is required and induces a Laplacian term in governing equations by solving the phonon Boltzmann equation. A general law containing the viscosity of the thermomass fluid is obtained in analogy with the Brinkman extension in porous hydrodynamics. The general law is then applied to investigate the effective thermal conductivity of a nanosystem, which covers the effect of the ultra-high heat flux and the boundary confinement and scattering. The present research not only presents a powerful heat conduction law for heat transport in nanosystems, but also bridges the microscopic Boltzmann transport equation and the macroscopic gas dynamics of phonons in terms of the thermomass theory.

In this paper we study the solution of a linear and nonlinear damped wave and dissipative wave equations by Adomian decomposition method. We illustrate that the analytic solutions and a reliable numerical approximation of the damped wave and dissipative wave equations are calculated in the form of a series with easily computable components. The nonhomogeneous problem is quickly solved by observing the self-canceling"noise"terms whose sum vanishes in the limit. In comparison to traditional techniques, the series based technique of Adomian decomposition method is shown to evaluate solutions accurately and cheaply.

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

Aims. We model helical or rotating signatures in the solar atmosphere to further understand the efficiency of the equilibrium conditions, for example magnetic twist, rotation, plasma-β, and viscous effects on the life of solar helical structures. Methods. Solar rotating structures, such as tornadoes, spirals, and whirls are modelled by considering a rotating and twisted magnetic cylinder residing in an environment with a straight magnetic field. A macroscopic approach proves adequate for working on the phase speed and damping of waves in solar atmospheric structures; as such, the magnetohydrodynamic theory is implemented. In this way the second order thin flux tube approximation is used for obtaining expressions for the frequency, deceleration, and damping of torsional waves in solar plasma structures in the presence of equilibrium rotation, magnetic twist, viscosity, and gravity. Results. The dependency of the dissipation effects regarding the torsional wave in the linear regime is highlighted. The dispersion relation for axisymmetric oscillations propagating along a rotating and twisted solar cylindrical plasma structure in the presence of plasma viscosity and gravity is obtained. In this way we present explicit expressions for the oscillation and damping of torsional waves. The explicit expressions shed light on the influence of the equilibrium and environmental conditions on the speed deceleration, frequency, and damping of the torsional wave that exists in various layers of the solar atmosphere. The dispersion of the torsional wave is highly controlled by the combined effects of the rotation and the plasma-β, where when both are zero, the magnetic twist becomes significant only when the plasma resistivity comes into play. Regarding damping, the dominant actor for coronal conditions is the magnetic twist. However, since the damping time is highly dependent on the plasma-β, for photospheric conditions, the rotation becomes very significant. The damping of torsional waves is inversely proportional to the elevation of the rotating structure. This means that if the torsional wave survives through the photosphere and chromosphere, the chance for it to extend through the corona and solar wind is very high by gradually dissipating energy, which gives more opportunity for it to be observed.

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular, our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations.

Dissipation-preserving spectral element method for damped seismic wave equations

In this work we focus on initial-boundary value problem of damped semilinear wave equation$u_{tt}-\Delta~u$ $+\,~u_t=|u|^p$ outside compact obstacles in high dimensions $(n\geq~3)$. We show thatthe solutions exist globally in time with power $1+\frac2n<p\leq\frac{n}{n-2}$, assuming that the initial data are small enough. The proof relies on a prior estimate of a weighted energy functional, established based on the decay estimates for linear damped wave equations.

Existence and nonexistence of time-global solutions to damped wave equation on half-line

The damped nonlinear wave equation, also known as the nonlinear telegraph equation, is studied within the framework of semigroups and eigenfunction approximation. The linear semigroup assumes a central role: it is bounded on the domain of its generator for all time This permits eigenfunction approximation within the semigroup framework, as a tool for the study of weak solutions. The semigroup convolution formula, known to be rigorous on the generator domain, is extended to the interpretation of weak solution on an arbitrary time interval. A separate approximation theory can be developed by using the invariance of the semigroup on eigenspaces of the Laplacian as the system evolves. For (locally) bounded continuous L 2 forcing, there is a natural derivation of a maximal solution, which can logically include a constraint on the solution as well. Operator forcing allows for the incorporation of concurrent physical processes. A significant feature of the proof in the nonlinear case is verification of successive approximation without standard fixed point analysis.