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.     

A two-dimensional inverse problem for an integro-differential equation of electrodynamics

An integro-differential equation corresponding to a two-dimensional problem of electrodynamics with dispersion is considered. It is assumed that the electrodynamic properties of a nonconducting medium with a constant magnetic permeability and the external current are independent of the x ₃ coordinate. In this case, the third component of the electric field vector satisfies a second-order scalar integro-differential equation with a variable permittivity of the medium. For this equation, we study the problem of finding the spatial part of the kernel entering the integral term. This corresponds to finding the part of the permittivity that depends on the electromagnetic frequency. It is assumed that the permittivity support is contained in some compact domain Ω ? ?². To find this coefficient inside Ω, we use information on the solution of the corresponding direct problem on the boundary of Ω on a finite time interval. An estimate for the conditional stability of the solution of the inverse problem is established under the assumption that the time interval is sufficiently large.     

Finding discontinuities in the coefficients of the linear nonstationary transport equations

An X-ray tomography problem that is an inverse problem for the transport differential equation is set up and investigated. The absorption and single scattering of particles are taken into account. The transport equation is nonstationary (its coefficients and the unknown function depend on time), involves multiple energy levels, and its coefficients can undergo jump discontinuities with respect to the spatial variable (in other words, the medium in which the process proceeds is inhomogeneous). The sought object is the set on which the coefficients of the equation suffer a discontinuity, which corresponds to the search for the boundaries between the different substances composing the sensed medium.     

Layerwise sensing in X-ray tomography in the polychromatic case

An X-ray tomography problem that is an inverse problem for the transport differential equation is studied. The absorption and single scattering of particles are taken into account. The suggested statement of the problem corresponds to stepwise and layerwise sensing of an unknown medium with initial data specified as the integrals of the outgoing flux density with respect to energy. The sought object is a set on which the coefficients of the equations suffer a discontinuity, which corresponds to searching for the boundaries between the different substances composing the sensed medium. A uniqueness theorem is proven under rather general assumptions and a condition guaranteeing the existence of the sought lines. The proof is constructive and can be used for developing a numerical algorithm.     

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.     

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.     

Optimal control problem with an integral equation as the control object

We consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints. This integral equation corresponds to the fractional moment of a stochastic process involving short-range and long-range dependences. For both cases, we derive the first-order necessary optimality conditions in the form of the Euler–Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection.     

A stability estimate for a solution to an inverse problem of electrodynamics

We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium. For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ³. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently large we estimate the conditional stability of the solution to this inverse problem.     

Self-Similar Parabolic Optical Solitary Waves

We study solutions of the nonlinear Schrödinger equation (NLSE) with gain, describing optical pulse propagation in an amplifying medium. We construct a semiclassical self-similar solution with a parabolic temporal variation that corresponds to the energy-containing core of the asymptotically propagating pulse in the amplifying medium. We match the self-similar core through Painlevé functions to the solution of the linearized equation that corresponds to the low-amplitude tails of the pulse. The analytic solution accurately reproduces the numerically calculated solution of the NLSE.     

Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media

A three-dimensional inverse scattering problem for the acoustic wave equation is studied. The task is to determine the density and acoustic impedance of a medium. A necessary and sufficient condition for the unique solvability of this problem is established in the form of an energy conservation law. The interpretation of the solution to the inverse problem and the construction of medium images are discussed.     

On the analysis of an endothermal/exothermal saturation problem

We consider an endo-or exo-thermal saturation problem that corresponds to a parabolic quasi-variational inequality. Applying regularity results and inequalities of Lewy-Stampacchia type, we prove the solvability of a modified problem (including the Steklov averaging and the mollification of the saturation velocity) for the nonlinear case and also of the exact problem for the linear case with a small coefficient in the temperature equation. Bibliography: 8 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 131–141.     

An Iterative Way of Solving the Inverse Scattering Problem for an Acoustic System of Equations in an Absorptive Layered Nonhomogeneous Medium

An algorithm is considered for solving the inverse scattering problem of seismic waves in a layered medium. The algorithm is based on solving a nonclassical ordinary differential equation with respect to an acoustic impedance, which also contains an unknown function characterizing the dissipative properties of the medium. The uniqueness of determining of these functions and the functional dependence associating them is established by solving the inverse problem of ground seismics. Results are presented from a computing experiment on applying the proposed algorithm.     

A free boundary problem in an absorbing porous material with saturation dependent permeability

We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.?The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules. Received December 1999     

Conservative finite-difference scheme for the problem of a femtosecond laser pulse with an axially symmetric profile propagating in a medium with cubic nonlinearity

Conservative finite-difference schemes are constructed for the problem of a femtosecond laser pulse propagating in a cubically nonlinear medium in the axially symmetric case with allowance for temporal dispersion of the nonlinear response of the medium. The process is governed by the nonlinear Schrödinger equation involving the time derivative of the nonlinear term. The invariants of the differential problem are presented. It is shown that the difference analogues of these invariants hold for the solution to the finite-difference schemes proposed for the problem. As an example, the numerical results obtained for the self-focusing of a femtosecond light beam are presented.     

On a Well-Posed Approximation of an Inverse Problem in Isotropic Medium with Memory

A problem of determining the function of memory from a model of longitudinal wave propagation in an isotropic medium is discussed. The external impulse is assumed to be close to a singular distribution. The problem is approximated by an equation of the second kind and an error estimate is provided. Errors of the method and initial data have a common order in the maximum norms, which implies that the problem is well-posed.     

The impact of a plane punch on an elastic half-plane

The transient dynamic contact problem of the impact of a plane absolutely rigid punch on an elastic half-plane is considered. The solution of the integral equation of this problem in terms of the unknown Laplace transform of the contact stresses at the punch base is constructed by a special method of successive approximations. The solution of the transient dynamic contact problem is obtained after applying an inverse Laplace transformation to the solution of the integral equation over the whole time range of the impact process, and the law of the penetration of the punch into the elastic medium is determined from a Volterra-type integrodifferential equation. The conditions for the punch to begin to separate from the elastic half-plane are formulated from the solution obtained, and all the stages of the separation process are investigated in detail. The law of the punch motion on the elastic half-plane and the width of the contact area, which varies during the separation, are then determined from the solution of the Volterra-type integrodifferential equation when an additional condition is satisfied.     

Numerical solutions of an extended Korteweg-de Vries equation describing solitary waves in turbulent open-channel flow

Solitary waves in two-dimensional, turbulent open-channel flow are considered. In an asymptotic analysis given in [1], assuming a bottom roughness that is varying along the channel bed, an extended Korteweg-de Vries (KdV) equation was derived to describe the surface elevation of the wave. We adopted this equation and solved it numerically as a coupled boundary-value eigenvalue problem, obtaining results for stationary and transient wave solutions as well as for the eigenvalue, which corresponds to distinct values of the bottom friction coefficient. While the numerical solutions as compared to the asymptotic solutions in [1] agree well in the stationary case, there were major differences found in the transient solutions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotationssymmetrische,wirbelbehaftete Strömung als Variationsproblem

Summary This is an analysis of the axially symmetric rotational flow which takes place (amongst other things) in turbomachinery. The problem can be reduced to a partial differential equation with given boundary conditons. Bearing in mind that the differential equation considered corresponds to a variational problem, the Ritz method is used for a numerical solutions. This requires a few additional assumptions to facilitate the computation procedure and enables us to correlate various design parameters. The velocity field being thus obtained, it is possible to determine the form of the blades.     

Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.     

Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play important roles for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.