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Abstract The ray statement of the inverse problem of determining three unknown variable coefficients in the linear acoustic equation is studied. These coefficients are assumed to differ from given constants only inside some bounded domain. There are point pulse sources and acoustic receivers on the boundary of this domain. Acoustic signals are measured by a receiver near the moment of time at which the signal from a source arrives at the receiver. It is shown that this information makes it possible to uniquely determine all the three desired coefficients. Algorithmically, the original inverse problem splits into three subproblems solved successively. One of them is a well-known inverse kinematic problem (of determining the speed of sound), while the other two lead to the same integral geometry problem for a family of geodesic lines determined by the speed of sound.

Abstract For the equation $${{u}_{{tt}}} - \Delta u - f(x,u) = 0, (x,t) \in {{\mathbb{R}}^{4}},$$ where $$f(x,u)$$ is a smooth function of its variables and is compact in x, the inverse problem of recovering this function from given information on solutions of Cauchy problems for the differential equation is studied. Plane waves with a strong front that propagate in a homogeneous medium in the direction of the unit vector ν and then impinge on an inhomogeneity localized inside some ball B(R) are considered. It is supposed that the solutions of the Cauchy problems can be measured on the boundary of this ball for all ν at times close to the arriving time of the front. The forward Cauchy problem is studied, and the existence of a unique bounded solution in a neighborhood of a characteristic wedge is stated. An amplitude formula for the derivative of the solution with respect to t on the front of the wave is derived. It is demonstrated that the solution of the inverse problem reduces to a series of X-ray tomography problems.

For a hyperbolic equation of the second order, we consider the inverse problem of recovering the coefficient $ q(x,y) $ in this equation. We discuss the scheme of solution of the problem which was proposed by Kabanikhin about 30 years ago. This scheme generalizes the Gelfand–Levitan–Krein method for the solution of the inverse spectral problem to the multidimensional case and reduces the solution of the inverse problem to some infinite system of linear integral equations. No mathematical justification for this scheme has been obtained yet. But numerical experiments based on the $ N $ -approximation produced good results. In this article, we justify some elements of the scheme related to the construction of the infinite system of integral equations in the case when the coefficient $ q(x,y) $ is analytic in $ x $ . In particular, we prove the convergence of the series in these equations and find the conditions for the $ N $ -approximation of the system. We also establish that the infinite system of integral equations is not Fredholm. The question of the solvability of the systems remains open.

An Erratum to this paper has been published: https://doi.org/10.1134/S106456242207002X

Given a hyperbolic equation with variable coefficients, we construct a regularizing algorithm to solve the problem of continuation of the wave field from the boundary of the half-plane inside it. We introduce some $$N$$ -approximate solutions and establish their convergence to the exact solution. Under consideration is the case when the problem data have an error of $$\delta $$ . We find an estimate of the accuracy of the approximate solutions and prove the convergence of the approximate solutions to the unique solution as $$\delta \to 0$$ .

For electrodynamic equations related to non-conducting dispersible medium we consider the inverse problem of recovering two variable coefficients from a given phaseless information of solutions to ...

For electrodynamic equations with permittivity specified by a symmetric matrix $$\varepsilon (x) = ({{\varepsilon }_{{ij}}}(x),i,j = 1,2,3)$$ , the inverse problem of determining this matrix from information on solutions of these equations is considered. It is assumed that the permittivity is a given positive constant $${{\varepsilon }_{0}} > 0$$ outside a bounded domain $$\Omega \subset {{\mathbb{R}}^{3}}$$ , while, inside $$\Omega $$ , it is an anisotropic quantity such that the differences $${{\varepsilon }_{{ij}}}(x) - {{\varepsilon }_{0}}{{\delta }_{{ij}}} = :{{\tilde {\varepsilon }}_{{ij}}}(x),$$ $$i,j = 1,2,3,$$ are small. Here, $${{\delta }_{{ij}}}$$ is the Kronecker delta. The inverse problem is studied in the linear approximation. The structure of the solution to a linearized direct problem for the electrodynamic equations is investigated, and it is proved that all elements of the matrix $$\tilde {\varepsilon }(x) = {{\tilde {\varepsilon }}_{{ij}}}(x),\;i,j = 1,2,3$$ , can be uniquely determined by special observation data. Moreover, the problem of recovering the diagonal components $${{\tilde {\varepsilon }}_{{ij}}}(x),\;i = 1,2,3,$$ leads to a usual X-ray tomography problem, so these components can be efficiently computed. The recovery of the other components leads to a more complicated algorithmic procedure.

A survey of recent research concerning phaseless inverse problems for several differential equations is given. Mainly, the surveyed studies were performed over the last five years, although their importance of this subject for quantum scattering theory was noted more than 40 years ago. Problem formulations and results are presented, and the basic ideas underlying the research are described.

Under study is the integral inequality that has as kernel a nonnegative polynomial in the powers of the difference of arguments and a large parameter N. We establish some inequality whose form agrees with the celebrated Gronwall-Bellman inequality in which the argument of the exponent depends linearly on N.

An Erratum to this paper has been published: https://doi.org/10.1134/S106456242234004X

For a system of electrodynamic equations, the inverse problem of determining an anisotropic conductivity is considered. It is supposed that the conductivity is described by a diagonal matrix σ(x) = $${\text{diag}}({{\sigma }_{1}}(x),{{\sigma }_{2}}(x)$$ , σ3(x)) with $$\sigma (x) = 0$$ outside of the domain Ω = $$\{ x \in {{\mathbb{R}}^{3}}|\left| x \right| < R\} $$ , $$R > 0$$ , and the permittivity e and the permeability μ of the medium are positive constants everywhere in $${{\mathbb{R}}^{3}}$$ . Plane waves coming from infinity and impinging on an inhomogeneity localized in Ω are considered. For the determination of the unknown functions $${{\sigma }_{1}}(x)$$ , $${{\sigma }_{2}}(x)$$ , and $${{\sigma }_{3}}(x)$$ , information related to the vector of electric intensity is given on the boundary S of the domain Ω. It is shown that this information reduces the inverse problem to three identical problems of X-ray tomography.

For the system of electrodynamics equations with anisotropic permittivity, the inverse problem of determining the permittivity is studied. It is supposed that the permittivity is characterized by a diagonal matrix $$\epsilon = {\text{diag}}({{\varepsilon }_{1}},{{\varepsilon }_{1}},{{\varepsilon }_{2}}),$$ where $${{\varepsilon }_{1}}$$ and $${{\varepsilon }_{2}}$$ are positive constants everywhere outside of a bounded domain $${{{\Omega }}_{0}} \subset {{\mathbb{R}}^{3}}$$. Time-periodic solutions of Maxwell’s equations related to two modes of plane waves coming from infinity and impinging on a local inhomogeneity located in $${{{\Omega }}_{0}}$$ are considered. For determining functions $${{\varepsilon }_{1}}(x)$$ and $${{\varepsilon }_{2}}(x),$$ some information on the magnitudes of the electric strength vectors of two interfering waves is given. It is shown that this information reduces the original problem to two inverse kinematic problems with incomplete data regarding travel times of electromagnetic waves. The linearized statement for these problems is investigated. It is shown that, in the linear approximation, the problem of determining $${{\varepsilon }_{1}}(x)$$ and $${{\varepsilon }_{2}}(x)$$ is reduced to two X-ray tomography problems.

The system of equations of electrodynamics is considered for a nonmagnetic nonconducting medium. For this system, the problem is under study of determining the permittivity e from the given modulus of the electric intensity vector of the electromagnetic field that is the result of the interference of the two fields generated by some point sources of an extraneous current. It is assumed that permittivity is different from a given positive constant e0 only inside a compact domain Ω0 ⊂ ℝ3, whereas the modulus of the electric field intensity vector is given for all frequencies starting from a fixed frequency ω0 on the boundary S of some domain Ω that includes Ω0. It is shown that this information allows us to reduce the original problem to the well-known inverse kinematic problem of determining the refraction index inside Ω by the travel time of the electromagnetic wave between the points of the boundary of this domain. The algorithm for numerical solution of the inverse problem is constructed, and the test calculations on the simulated data are presented.

For nonmagnetic and nonconductive medium the system of electrodynamic equations that corresponds to periodic in time oscillations is considered. An inverse problem of determining permittivity in this system by the given module of the electric strength is studied. It is supposed that the electric fields is a result of the interference of two fields created by point sources. The permittivity e(x) is assumed to be differ from a given positive constant e0 inside of a compact domain W0 Ì R3 only. An information on module of the electric strength is given on the boundary of the domain W contained W0 inside itself and for all frequencies beginning with some fixed frequency w0. The asymptotic behavior of solution of a direct problem related to the electrodynamic equations is studied and the original inverse problem is reduced to the well known inverse kinematic problem. This reduction open a way for constructive solution of the inverse phaseless problem.

For an unmagnetized nonconducting medium, a system of electrodynamic equations corresponding to time-periodic oscillations is considered. For this system, we study the problem of determining the permittivity $$\varepsilon $$ from a given electric field magnitude in the electromagnetic field resulting from the interference of two fields produced by external point current sources. It is assumed that the permittivity differs from a given positive constant $${{\varepsilon }_{0}}$$ only inside a compact domain $${{\Omega }_{0}} \subset {{\mathbb{R}}^{3}}$$ and the electric field magnitude is given for all frequencies, starting at a fixed one $${{\omega }_{0}}$$ , on the boundary $$S$$ of a domain $$\Omega $$ surrounding $${{\Omega }_{0}}$$ . By analyzing an asymptotic expansion of the solution to a direct problem at high frequencies, it is shown that, with the use of given information, the original problem is reduced to the well-known inverse kinematic problem of determining the refraction coefficient inside $$\Omega $$ from the travel times of an electromagnetic wave propagating between boundary points of this domain. This reduction opens up the possibility of obtaining a constructive solution of the original problem.

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.

Under consideration is the problem of continuation of the wave field from the boundary of a half-plane inside it. We obtain stability estimate for the solution of the corresponding Cauchy problem.

A regularizing algorithm is constructed for solving the problem of extension of a wave field from the boundary to the interior of a half-plane. The cases are considered of the wave equation and hyperbolic equations with variable lower order coefficients.

We consider the inverse problems for differential equations with complex-valued solutions in which the modulus of a solution to the direct problem on some special sets is a given information in order to determine coefficients of this equation; the phase of this solution is assumed unknown. Earlier, in similar problems the modulus of the part of a solution that corresponds to the field scattered on inhomogeneities in a wide range of frequencies was assumed given. The study of high-frequency asymptotics of this field allows us to extract from this information some geometric characteristics of an unknown coefficient (integrals over straight lines in the problems of recovering the potential and Riemannian distances between the boundary points in the problem of the refraction index recovering). But this is physically much more difficult to measure the modulus of a scattered field than that of the full field. In this connection the question arises how to state inverse problems with the full-field measurements as a useful information. The present article is devoted to the study of this question. We propose to take two plane waves moving in opposite directions as an initiating field and to measure the modulus of a full-field solution relating to interference of the incident waves. We consider also the problems of recovering the potential for the Schrodinger equation and the permittivity coefficient of the Maxwell system of equations corresponding to time-periodic electromagnetic oscillations. For these problems we establish uniqueness theorems for solutions. The problems are reduced to solving some well-known problems.

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient e from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant e0 only inside some compact domain Ω ⊂ R3 with smooth boundary S. To recover e inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω 0 on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.

The stationary system of Maxwell equations for a unmagnetized nonconducting medium is considered. For this system, the problem of determining the permittivity e from given electric or magnetic fields is studied. It is assumed that the electromagnetic field is induced by a plane wave coming from infinity in the direction ν. It is also assumed that the permittivity is different from a given positive constant e0 only inside a compact domain Ω ⊂ R 3 with a smooth boundary S. To find e inside Ω, the solution of the corresponding direct problem for the system of electrodynamic equations on the shadow portion of the boundary of Ω is specified for all frequencies starting at some fixed ω0 and for all ν. The high-frequency asymptotics of the solution to the direct problem is studied. It is shown that the information specified makes it possible to reduce the original problem to the well-known inverse kinematic problem of determining the refraction coefficient inside Ω from the traveling times of an electromagnetic wave. This leads to a uniqueness theorem for the solution of the problem under consideration and opens up the opportunity of its constructive solution.

The Cauchy problem for a linear second-order uniformly parabolic equation with variable coefficients depending only on the spatial variable is considered. For two- and three-dimensional spaces, asymptotic expansions of the solution as t → +0 are written. It is shown how these expansions can be used in the study of inverse problems of determining the coefficients of a parabolic equation.

We consider the problem of finding the boundaries of discontinuities and jumps at these discontinuities of a piecewise smooth function for X-ray tomography. We give some algorithm for the reconstruction of the unknown values as well as formulas for their calculation. The original problem of X-ray tomography is reduced to a problem for a continuous piecewise differentiable function.

For the integrodifferential viscoelasticity equations, we study the problem of determining the coefficients of the equations and the kernels occurring in the integral terms of the system of equations. The density of the medium is assumed to be given. We suppose that the inhomogeneity support of the sought functions is included in some compact domain B 0. We consider a series of inverse problems in which an impulse source is concentrated at the points y of the boundary of B 0. The point y is the parameter of the problem. The given information about the solution is the trace of the solution to the Cauchy problem with zero initial data. This trace is given on the boundary of B 0 for all y ∈ ∂B 0 and for a finite time interval. The main result of the article consists in obtaining uniqueness theorems for a solution to the initial inverse problem.

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ℝ2 | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.

Under study is the problem of finding the kernel and the index of dielectric permeability for the system of integrodifferential electrodynamics equations with wave dispersion. We consider a direct problem in which the external pulse current is a dipole located at a point y on the boundary ∂B of the unit ball B. The point y runs over the whole boundary and is a parameter of the problem. The information available about the solution to the direct problem is the trace on ∂B of the solution to the Cauchy problem given for the times close to the time when a wave from the dipole source arrives at a point x. The main result of the article consists in obtaining some theorems related to the uniqueness problems for a solution to the inverse problem.

For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration.

We consider a model of linear acoustics describing diffraction of an acoustic wave on a local anisotropic inhomogeneity. Under study is the existence of inhomogeneities which do not scatter the incident acoustic fields that are generated by any exterior compactly supported sources. We prove the existence of the above-mentioned inhomogeneities. Some constructive way is given of constructing a class of anisotropic media depending on an arbitrary function of one variable. We obtain some explicit formulas that define the main parameters of the shell.

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.

Let $\mathcal B$ be a viscoelastic body with a (smooth) bounded open reference set $\Omega$ in $\mathbb R^3$, with the equation of motion being described by the Lame coefficients $\lambda_0$ and $\mu_0$ and the related viscoelastic coefficients $\lambda_1$ and $\mu_1$. The latter are assumed to be factorized with the same temporal part, i.e. $\lambda_1(t,x)=k(t)p(x)$ and $\mu_1(t,x)=k(t)q(x)$. Furthermore, it is assumed that the spatial parts $p$ and $q$ of $\lambda_1$ and $\mu_1$ are unknown and the three additional measurements $\sum_{j=1}^3\sigma_{i,j}^0(t,x)$n$_j(x) = g_i(t,x)$, $i=1,2,3$, are available on $(0,T)\times \partial \Omega$ for some (sufficiently large) subset $\Gamma\subset \partial \Omega$. The fundamental task of this paper is to show the uniqueness of the pair $(p,q)$ as well as its continuous dependence on the boundary conditions, the initial data being kept fixed and the initial velocity being suitably related to the initial displacement.

The review summarizes current information on the biological activity and search of the antineoplastic mechanism of action among alkyl glycerolipids. Special attention is paid to following problems considered in experimental papers published during the last decade: selective ability phosphorus alkyl glycerolipids towards tumor cells using edelfosine and its analogues. The review contains modern information available in the literature, on a possible mechanism cytotoxic effect of such compounds.

We consider the viscoelasticity equation and the problem of recovering the spatial part p(x) of the Lame kernel depending on two spatial variables. We obtain a stability estimate of the solution to this problem.

The review is summarizes current information on biological activity and search of the antineoplastic mechanism of action of alkyl glycerolipids. Special attention is paid to following problems: selective ability phosphorus alkyl glycerolipids, antineoplastic activity, cytostatic and cytotoxic effects of edelfosine and its analogues. The review contains set of the data known for today from the literature, on the possible mechanism cytoyoxic actions of such connections.

We consider the problem of constructing a fundamental solution to a second order hyperbolic linear equation with variable coefficients depending on the space variable x ∈ ℝn. Under the assumption of high but finite smoothness of the coefficients, we write out the structure of a fundamental solution, establish smoothness of the coefficients of the expansion of its singular part, and characterize smoothness of the regular part.