The statement and numerical solution of an optimization problem in X-ray tomography

A nonclassical problem is considered for the transport equation with coefficients depending on the energy of radiation. The task is to find the discontinuity surfaces for the coefficients of the equation from measurements of the radiation flux leaving the medium. For this tomography problem, an optimization problem is stated and numerically analyzed. The latter consists in determining the radiation energy that ensures the best reconstruction of the unknown medium. A simplified optimization problem is solved analytically.     

High-order one-way model nesting in dispersive non-uniform media

Global-regional model interaction is considered for two-dimensional linear time dependent waves in a dispersive non-uniform medium with a continuously varying wave speed. The setup, which is sometimes called ‘one-way nesting,’ arises in Numerical Weather Prediction (NWP) as well as in other fields concerning waves in very large domains. The Carpenter scheme for this type of problem is revisited, in the context of the dispersive wave equation with a variable wave speed. The original Carpenter scheme is based on the Sommerfeld radiation operator, and thus is associated with low-order accuracy. By replacing the Sommerfeld operator with the high-order Hagstrom-Warburton absorbing operator, a modified Carpenter open boundary condition emerges which possesses high-order accuracy. This is demonstrated via a numerical example in a wave guide with a wave speed which varies linearly in the cross section.     

Traveling waves for a dissipative modified KdV equation

In this paper we consider a dispersive–dissipative nonlinear equation which can be regarded as a dissipation perturbed modified KdV equation, governing the evolution of long waves in an elastic rod immersed inside a viscoelastic medium. Using geometric singular perturbation theory, a construction of traveling waves for the equation is shown. This also is illustrated by presenting some numerical calculations.     

Finite element study of time‐dependent Maxwell's equations in dispersive media

In this article, we consider the time‐dependent Maxwell's equations in a bounded domain when dispersive media are involved. The Crank‐Nicolson scheme is developed to approximate the electric field equation by Nedelec edge elements and is proved to be optimal convergent in energy norm. The analysis is carried out for Debye medium, but the same results hold true for other dispersive media such as plasma and Lorentz medium. Furthermore, our analysis extends straightforward to cases when a dispersive medium and a simple medium (such as air) are coupled. Mathematics Subject Classification (2000): 65N30, 35L15, 78‐08. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Finite‐difference time‐domain simulation of acoustic propagation in heterogeneous dispersive medium

Accurate modeling of pulse propagation and scattering is a problem in many disciplines (i.e., electromagnetics and acoustics). It is even more tenuous when the medium is dispersive. Blackstock [D. T. Blackstock, J Acoust Soc Am 77 (1985) 2050] first proposed a theory that resulted in adding an additional term (the derivative of the convolution between the causal time‐domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium. Thus deriving a modified wave equation applicable to either linear or nonlinear propagation. For the case of an acoustic wave propagating in a two‐dimensional heterogeneous dispersive medium, a finite‐difference time‐domain representation of the modified linear wave equation can been used to solve for the acoustic pressure. The method is applied to the case of scattering from and propagating through a 2‐D infinitely long cylinder with the properties of fat tissue encapsulating a cyst. It is found that ignoring the heterogeneity in the medium can lead to significant error in the propagated/scattered field. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Extensional and flexural modes of Rayleigh–Lamb wave in an orthotropic thermoelastic layer lying over a viscoelastic half-space

In this article the characteristics of the extensional and flexural modes, propagating in a thermoelastic orthotropic layer lying over a viscoelastic half-space, are analyzed. The complete analysis is carried out in the framework of a thermodynamically consistent hyperbolic type heat conduction model without energy dissipation. The normal-mode-analysis is adopted and a general form of dispersive equation is derived for an anisotropic thermoelastic layered medium. A prominent distinction with the isotropic elastic solids is observed in the symmetric as well as anti-symmetric modes of dispersion curves. In turn, such deformation reshapes the wave propagation while the deformation stiffening changes significantly the phase velocities of the wave till the acoustic radiation stresses are balanced by elastic stresses in the current configuration of the hyperelastic medium.     

Error estimate for a fully discrete spectral scheme for Korteweg-de Vries-Kawahara equation

We are concerned with convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (Kawahara equation, in short), which is a transport equation perturbed by dispersive terms of the 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier-Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L ²-error bound of spectral accuracy in space and of second-order accuracy in time.     

The Kinetic Transport Equation in the Case of Compton Scattering

We improve the well-known form of the transport equation accounting for Compton scattering. We pose and study the direct problem of finding the radiation density distribution for given characteristics of a medium and known density of exterior sources. We prove existence and uniqueness theorems for a solution to the boundary value problem under consideration. The character of constraints corresponds mostly to the process of photon migration in a substance whose characteristics vary continuously with the space and energy variables. Unlike similar results, the assertions are proven without using the traditional inequalities for the coefficients of the transport equation.     

What is sustainable development? An attempt to interpret it as a soliton-like phenomenon

An attempt is made to establish a relation between the question of what ‘sustainable development’ means and the non-linear theory of shock waves. Despite the presence of dispersive, i.e. entropy-producing, forces a soliton-like, isentropic, transport of a wilfully desired distribution in a field of traded commodities is possible. Starting with the classical Korteweg de Vries (KdV) equation, two other examples, a sigmoidal and a Gaussian soliton in a diffusional environment, are analyzed in detail as a guide-line of how a ‘sustainable’ transport of an economically defined creation can be carried through time.     

A polychromatic inhomogeneity indicator in an unknown medium for an X-ray tomography problem

We pose and study an X-ray tomography problem, which is an inverse problem for the transport differential equation, making account for particle absorption by a medium and single scattering. The statement of the problem corresponds to a stage-by-stage probing of the unknown medium common in practice. Another step towards a more realistic problem is the use of integrals over energy of the density of emanating radiation flux as the known data, in contrast to specifying the flux density for every energy level, as it is customary in tomography. The required objects are the discontinuity surfaces of the coefficients of the equation, which corresponds to searching for the boundaries between various substances contained in the medium. We prove a uniqueness theorem for the solution under quite general assumptions and a condition ensuring the existence of the required surfaces. The proof is rather constructive in character and suitable for creating a numerical algorithm.     

Kinetic theory of a super-radiating laser

Statistical properties of the radiation of a super-radiating laser are studied on the basis of the kinetic equation for the density matrix of the generating field. An original derivation of this equation is proposed. Statistical properties of the laser medium are investigated in the stationary generating regime. We analyze the role played by the decay rate of intracavity photon fluctuations and by the statistical properties of the active medium in the suppression of the photocurrent shot noise. We consider the external pump to be weak, which is physically the most interesting case. It is shown that if the active medium is weakly excited, then only a 50% depression of the shot noise is possible. In the case of a strongly excited medium, depression can reach 100%. The spectrum of optical generation is obtained for conditions far from the instability zone.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 250–269, November, 1996.     

Radiative transfer in a two-dimensional rectangular annulus medium

The radiative transfer equation in a two-dimensional rectangular annulus medium is solved numerically. The numerical method is based on a finite difference scheme and a product quadrature discrete-ordinate scheme. The discretized equation of transfer is solved iteratively to give the radiation intensity. The medium is assumed to absorb, emit, and anisotropically scatter radiation. It is exposed to diffusely emitting and diffusely reflecting boundaries. The results of the total intensity for various radiative parameters are presented. The method can be modified easily to solve the rectangular medium without the annulus. Our results in this case compare very well with those of Crosbie et al. [1], Thynell et al. [2], and Wu [3].     

The evolution of nonlinear standing waves in bounded media

Summary The evolution of small amplitude disturbances in a bounded medium, under fixed and nearly fixed end conditions, is considered. The various physical effects accounted for are amplitude dispersion, frequency dispersion and dissipation due to both radiation of energy out of the medium and rate-dependence of the medium. In a nonlinear geometrical acoustics theory the transport equations which determine the signal carried by a component wave have the form of a simple wave equation, Korteweg-de Vries equation, damped simple wave equation and Burgers' equation.

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Zusammenfassung Die Entwicklung von Störungen kleiner Amplitude in einem begrenzten Medium wird untersucht, mit festen und nahezu festen Endbedingungen. Die berücksichtigten physikalischen Effekte sind Amplituden-Dispersion, Frequenz-Dispersion und Dissipation sowohl durch Abstrahlung von Energie aus dem Medium wie auch durch die Deformationsgeschwindigkeit im Medium. In der nicht-linearen geometrischen Akustik ist die Transportgleichung, welche das von einer Wellenkomponente übertragene Signal bestimmt, die einfache Wellengleichung, bezw. die Korteweg-de Vries-Gleichung, die gedämpfte einfache Wellengleichung und die Burgers-Gleichung.

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Presented at EUROMECH 73, Aix-en-Provence, April 1976.     

On a model in radiation hydrodynamics

We consider a simplified model arising in radiation hydrodynamics based on the Navier–Stokes–Fourier system describing the macroscopic fluid motion, and a transport equation modeling the propagation of radiative intensity. We establish global-in-time existence for the associated initial–boundary value problem in the framework of weak solutions.     

On a mathematical model for laser-induced thermotherapy

We study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well.     

Thermal radiation effects on magnetohydrodynamic free convection heat and mass transfer from a sphere in a variable porosity regime

A mathematical model is presented for multiphysical transport of an optically-dense, electrically-conducting fluid along a permeable isothermal sphere embedded in a variable-porosity medium. A constant, static, magnetic field is applied transverse to the cylinder surface. The non-Darcy effects are simulated via second order Forchheimer drag force term in the momentum boundary layer equation. The surface of the sphere is maintained at a constant temperature and concentration and is permeable, i.e. transpiration into and from the boundary layer regime is possible. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite difference scheme. Increasing porosity (ε) is found to elevate velocities, i.e. accelerate the flow but decrease temperatures, i.e. cool the boundary layer regime. Increasing Forchheimer inertial drag parameter (Λ) retards the flow considerably but enhances temperatures. Increasing Darcy number accelerates the flow due to a corresponding rise in permeability of the regime and concomitant decrease in Darcian impedance. Thermal radiation is seen to reduce both velocity and temperature in the boundary layer. Local Nusselt number is also found to be enhanced with increasing both porosity and radiation parameters.     

Terminal Damping of a Solitary Wave Due to Radiation in Rotational Systems

The evolution of a solitary wave under the action of rotation is considered within the framework of the rotation-modified Korteweg–de Vries equation. Using an asymptotic procedure, the solitary wave is shown to be damped due to radiation of a dispersive wave train propagating with the same phase velocity as the solitary wave. Such a synchronism is possible because of the presence of rotational dispersion. The law of damping is found to be "terminal" in the sense that the solitary wave disappears in a finite time. The radiated wave amplitude and the structure of the radiated "tail" in space–time are also found. Some numerical results, which confirm the approximate theory developed here, are given.     

Nonstandard methods for the convective‐dispersive transport equation with nonlinear reactions

A new nonstandard Eulerian‐Lagrangian method is constructed for the one‐dimensional, transient convective‐dispersive transport equation with nonlinear reaction terms. An “exact” difference scheme is applied to the convection‐reaction part of the equation to produce a semi‐discrete approximation with zero local truncation errors with respect to time. The spatial derivatives involved in the remaining dispersion term are then approximated using standard numerical methods. This approach leads to significant, qualitative improvements in the behavior of the numerical solution. It suppresses the numerical instabilities that arise from the incorrect modeling of derivatives and nonlinear reaction terms. Numerical experiments demonstrate the scheme's ability to model convection‐dominated, reactive transport problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 617–624, 1999