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We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in  Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in  Ω , u ( x , t ) = 0 , in  ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ), N > 2, p ∈ ( 2 , 2 s ∗ ], μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions. [ABSTRACT FROM AUTHOR]

The author considers inverse problems for a hyperbolic equation with a singular perturbation, in which the unknown function is one involved in the source term. The existence of a solution to the inverse problems is proved, numerical ways of solving them are developed, and results are presented from computational experiments illustrating their efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we present the L 2 -norm stability analysis and error estimate for the explicit single-step time-marching discontinuous Galerkin (DG) methods with stage-dependent numerical flux parameters, when solving a linear constant-coefficient hyperbolic equation in one dimension. Two well-known examples of this method include the Runge–Kutta DG method with the downwind treatment for the negative time marching coefficients, as well as the Lax–Wendroff DG method with arbitrary numerical flux parameters to deal with the auxiliary variables. The stability analysis framework is an extension and an application of the matrix transferring process based on the temporal differences of stage solutions, and a new concept, named as the averaged numerical flux parameter, is proposed to reveal the essential upwind mechanism in the fully discrete status. Distinguished from the traditional analysis, we have to present a novel way to obtain the optimal error estimate in both space and time. The main tool is a series of space–time approximation functions for a given spatial function, which preserve the local structure of the fully discrete schemes and the balance of exact evolution under the control of the partial differential equation. Finally some numerical experiments are given to validate the theoretical results proposed in this paper. [ABSTRACT FROM AUTHOR]

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size HH and the fine grid size hh satisfy h=O(H(2k+1)⁄(k+1))h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) (k≥1k\ge 1), where kk is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

We consider the problem in a cylindrical domain of variables t and x , which is the Cartesian product of a segment containing zero and a unit circle and is split into two subdomains by a hyperplane t = 0. In each subdomain, the solution of the problem satisfies the corresponding differential equations with multipoint conditions in the case of multiple nodes and the conditions of linear conjugation on the interface t = 0. The form of the domain imposes additional conditions on the periodicity of the solution with respect to the space variable x. We study the conditions of correct solvability of the problem in the Sobolev space closely related to the problem of small denominators and their estimation. With the help of the metric approach, it is shown that these conditions are satisfied for almost all (with respect to the Lebesgue measure) vectors formed by the interpolation nodes of the multipoint conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the blow-up of solutions for hyperbolic equations involving the fractional Laplacian operator with damping and source terms. We obtain the global existence results. Then, we observe the blow-up of solutions using the concavity method. Finally, we present some numerical simulation results. [ABSTRACT FROM AUTHOR]

In the half-plane, a solution to a two-dimensional hyperbolic equation with a translation operator in the lowest derivative with respect to a spatial variable varying along the entire real axis is constructed in an explicit form. It is proven that the solutions obtained are classical if the real part of the symbol of a differential-difference operator in the equation is positive. [ABSTRACT FROM AUTHOR]

With advancements in mineral processing technology, the disposal of fine-grained tailings has increasingly become a significant challenge. The geotextile tube method, characterized by its use of a permeable fabric and its cost-effectiveness, has gradually been applied in dam construction and other engineering projects involving tailings. This method offers a novel approach to addressing the storage issues of fine-grained tailings and promotes sustainable utilization. In this paper, the fine tailings that remained after the cyclone classification of Ganzhou tungsten ore were taken as the research object. Specifically, this research endeavored to evaluate the effects of various filling heights and concentrations on the geotextile tube-filling and consolidation process. The results revealed that the filling concentration had a significant impact on the filling benefit of the geotextile tubes, while the filling height had a minimal effect. During the consolidation drainage stage, the dry density, internal friction angle, cohesion, and compression modulus of the tailings in the bags increased with an increasing consolidation time and filling concentration. However, the physical and mechanical properties of the tailings in the geotextile tubes decreased with an increased filling height. Ultimately, this research developed a hyperbolic equation that makes it possible to forecast the ultimate settlement value at various filling heights and concentrations, better representing how the settlement of geotextile tubes changes over the consolidation time. [ABSTRACT FROM AUTHOR]

We study a hyperbolic equation with an arbitrary number of potentials undergoing translation in arbitrary directions. Differential-difference equations arise in various applications that are not covered by the classical theory of differential equations. In addition, they are of considerable interest from a theoretical point of view, since the nonlocal nature of such equations gives rise to various effects that do not arise in the classical case. We find a condition on the vector of coefficients for nonlocal terms in the equation and on the vectors of potential translations that ensures the global solvability of the equation under consideration. By imposing the specified condition on the equation and using the classical Gelfand–Shilov scheme, we explicitly construct a three-parameter family of smooth global solutions to the equation under study. [ABSTRACT FROM AUTHOR]

Starting from a nonlocal version of a classical kinetic traffic model, we derive a class of second-order macroscopic traffic flow models using appropriate moment closure approaches. Under mild assumptions on the closure, we prove that the resulting macroscopic equations fulfill a set of conditions including hyperbolicity, physically reasonable invariant domains and physically reasonable bounds on the speed with which the waves propagate. Finally, numerical results for various situations are presented, illustrating the analytical findings and comparing kinetic and macroscopic solutions. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a one-dimensional hyperbolic equation whose lower-order coefficient and right-hand side oscillate in time with a high frequency and the amplitude of the lower-order coefficient is small. Under study is the reconstruction of the cofactors of these rapidly oscillating functions independent of the space variable from a partial asymptotics of a solution at some point of the space. The classical theory of inverse problems examines the numerous problems of determining unknown sources, and coefficients without rapid oscillations for various evolutionary equations, where the exact solution to the direct problem appears in the additional overdetermination condition. Equations with rapidly oscillating data are often encountered in modeling the physical, chemical, and other processes that occur in media subjected to high-frequency electromagnetic, acoustic, vibrational, and others fields, which demonstrates the topicality of perturbation theory problems on the reconstruction of unknown functions in high-frequency equations. We give some nonclassical algorithm for solving such problems that lies at the junction of asymptotic methods and inverse problems. In this case the overdetermination condition involves a partial asymptotics of solution of a certain length rather than the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some of important qualitative properties of solutions of secondorder hyperbolic equations, whose coefficients of the terms involving the second-order derivatives are independent of the desired function and its derivatives. Solutions of these equations have a special property called curvilinear parallelogram identity (or mean-value property), which can be used to solve some initial-boundary value problems. [ABSTRACT FROM AUTHOR]

The paper is devoted to constructing a numerical algorithm for calculating the blood flow in a volume vessel. A system of differential equations describing the dynamics of fluid in a single vessel with moving walls in cylindrical coordinates is derived assuming axial symmetry in arbitrary Eulerian-Lagrangian variables. A balance-characteristic scheme based on the Cabaret methodology is constructed for the obtained system of equations. The results of calculations of test problems are given. [ABSTRACT FROM AUTHOR]

We study the multidimensional analogue of the Lavrent'ev integral equation to which an inverse problem of acoustic sounding is reduced. Conditions under which the studied equation has a unique solution are established. Results of numerical experiments concerning the solution of the inverse acoustic problem with variously located sets of sources and detectors are presented. [ABSTRACT FROM AUTHOR]

In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution to some integral equation. The local solvability of this euqation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray–Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established. [ABSTRACT FROM AUTHOR]

For a linear hyperbolic equation of fourth order, a Dirichlet type boundary problem in an orthogonally convex domain is investigated. Sharp sufficient conditions guaranteeing solvability and well-posedness of the problem under consideration are established. [ABSTRACT FROM AUTHOR]

In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem. [ABSTRACT FROM AUTHOR]

In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem.

The authors analyze continuity equations with Stratonovich stochasticity, defined on a smooth closed Riemannian manifold M with metric h. The velocity field u is perturbed by Gaussian noise terms Ẇ1(t), ..., ẆN(t) driven by smooth spatially dependent vector fields a1(x), ..., aN(x) on M. The velocity u belongs to Lt1Wx1,2 with divhu bounded in Lt,xp for p > d + 2, where d is the dimension of M (they do not assume divhu ∈ Lt,x∞). For carefully chosen noise vector fields ai (and the number N of them), they show that the initial-value problem is well-posed in the class of weak L2 solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this "regularization by noise" result is based on a L2 estimate, which is obtained by a duality method, and a weak compactness argument. [ABSTRACT FROM AUTHOR]

Various models of membrane oscillations emerging in the theory of elasticity of mechanical systems, biomechanics of the internal ear of vertebrata, and in the theory of electrical circuits are discussed in the article. The considered oscillations have different natures, but their mathematical models are described using similar initial boundary value problems for the second-order hyperbolic equation with the nontrivial boundary condition. The differential equations in these problems are the same. Thus, for example, the model of voltage distribution in the telegraph line emerges for the one-dimensional equation of oscillations. The model of oscillations of a circular homogeneous solid membrane, a membrane with a hole, and the model of gas oscillations in a sphere and spherical region emerge for the two-dimensional and three-dimensional operators, but take into account the radial symmetry of oscillations. The model problem on membrane oscillation can be considered as the problem on ear drum membrane oscillations. The unified approach to reducing the corresponding problems to the initial boundary value problem with zero boundary conditions is suggested. The technique of formulating the solution in the form of a Fourier series using eigenfunctions of the corresponding Sturm–Liouville problem is described. [ABSTRACT FROM AUTHOR]

In the half-plane, a solution to a two-dimensional hyperbolic equation with a translation operator in the lowest derivative with respect to a spatial variable varying along the entire real axis is constructed in an explicit form. It is proven that the solutions obtained are classical if the real part of the symbol of a differential-difference operator in the equation is positive.

An initial boundary value problem for an inhomogeneous second-order hyperbolic equation on a finite segment with constant coefficients and a mixed derivative is investigated. The case of fixed ends is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. The classical solution of the initial boundary value problem is determined. The uniqueness theorem of the classical solution is formulated and proved. A formula is given for the solution in the form of a series whose members are contour integrals containing the initial data of the problem. The corresponding spectral problem for a quadratic beam is constructed and a theorem is formulated on the expansion of the first component of a vector-function with respect to the derivative chains corresponding to the eigenfunctions of the beam. This theorem is essentially used in proving the uniqueness theorem for the classical solution of the initial boundary value problem.

In this research work, we proposed a Haar wavelet collocation method (HWCM) for the numerical solution of first- and second-order nonlinear hyperbolic equations. The time derivative in the governing equations is approximated by a finite difference. The nonlinear hyperbolic equation is converted into its full algebraic form once the space derivatives are replaced by the finite Haar series. Convergence analysis is performed both in space and time, where the computational results follow the theoretical statements of convergence. Many test problems with different nonlinear terms are presented to verify the accuracy, capability, and convergence of the proposed method for the first- and second-order nonlinear hyperbolic equations.

In this paper we study the behaviour of solutions for a nonlocal hyperbolic equation. We use the Pohozaev manifold combined with a new technique to explicit two invariant regions in the space of initial data. On the first one the solution blows up (in finite or infinite time) and in the second one the solution exists globally. Additionally, we prove that the ground state solution of the elliptic problem associated to the original problem is unstable. The main goal of this paper is to present a new technique which allows us to consider nonlocal problems and to extend the classical result proved by Shatah (Trans Am Math Soc 290(2):701–710, 1985). [ABSTRACT FROM AUTHOR]

A -hyperbolic equation with a Bessel operator in time and with an operator in space is considered. The operator includes singular differential Bessel operators with non-negative parameters and derivatives of these operators. The found fundamental solution is expressed in terms of the Bessel function of the first kind of negative order. [ABSTRACT FROM AUTHOR]

In this paper, we study the question of constructing explicit solutions in a half-space of a hyperbolic equation containing translation operators in space variables in all coordinate directions. Such equations are a natural generalization of classical equations of hyperbolic type, and the resulting solution relates the value of the desired function at different points of the half-space where the process takes place. To construct solutions, a classical operating scheme is used, namely, the formal application of an integral transformation. A theorem is proved that the constructed solutions are classical if the real part of the symbol of the differential-difference operator in the equation is positive. Classes of equations for which this condition is satisfied are given. [ABSTRACT FROM AUTHOR]

In this paper we consider small oscillations of the ideal gas near its equilibrium state inside an unbounded cylindrical tube. We investigate radial oscillations of the gas within a fixed cross-section. The gas is radially inhomogeneous. There is a power dependence of the gas density on the radial coordinate. And also the Boyle–Mariotte law is fulfilled at a constant temperature. We study the pressure in a gas, the values of which are known at the initial and final times of the experiment, and the average value of the pressure at any other time is constant. This physical problem is modeled by a non-local boundary value problem with an integral condition for a hyperbolic equation with a singular coefficient in a rectangular domain. Uniqueness and existence theorems for a solution to the problem are proved. The solution of the problem is constructed in an explicit analytical form. [ABSTRACT FROM AUTHOR]

In this paper, a quasi-interpolation method is proposed for solving hyperbolic conservation laws based on the essentially non-oscillatory (ENO) scheme. The hyperbolic equation is discretized in space with the finite difference ENO method and then the semi-discrete system is integrated by the strong stability preserving Runge–Kutta scheme. However, in the step of ENO reconstruction, the finite difference method is replaced by several quasi-interpolation schemes, including the multiquadric quasi-interpolation, the integral-type multiquadric quasi-interpolation and the cubic B-spline quasi-interpolation. Our quasi-interpolation method is simple and easy to implement since it doesn't need to solve any linear system. Moreover, it is also suitable for nonuniform grids and noisy sampling data. Nonlinear hyperbolic problems we target include one-dimensional and two-dimensional Burger equations and also the Euler equation. Numerical results demonstrate that the proposed quasi-interpolation ENO method is stable and has good accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior (as ) of solutions to the initial boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis. The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum of the one-dimensional Schrödinger operator , when the left end of the spectrum of this operator is negative. [ABSTRACT FROM AUTHOR]

Using an operational scheme, a three-parameter family of solutions is constructed in a half-space for a multidimensional hyperbolic differential-difference equation with translation operators of the general type acting on all spatial variables. The theorem is proved stating that the obtained solutions are classical, provided that the real part of the symbol of the differential-difference operator is positive. Classes of equations are given for which the indicated condition is satisfied. [ABSTRACT FROM AUTHOR]

A definition and justify a method for constructing the Green’s function of the second Darboux problem for a two-dimensional linear hyperbolic equation of the second order in a characteristic triangle is given. In contrast to the (well-developed) theory of the Green’s function for self-adjoint elliptic problems, this theory has not yet been developed. And for the case of asymmetric boundary value problems such studies have not been carried out. It is shown that the Green’s function for a hyperbolic equation of the general form can be constructed using the Riemann-Green function for some auxiliary hyperbolic equation. The notion of the Green’s function is more completely developed for Sturm-Liouville problems for an ordinary differential equation, for Dirichlet boundary value problems for Poisson equation, for initial boundary value problems for a heat equation. For many particular cases, the Greens’ function has been constructed explicitly. However, many more problems require their consideration. In this paper, the problem of constructing the Green’s function of the second Darboux problem for a hyperbolic equation was investigated. The Green’s function for the hyperbolic problems differs significantly from the Green’s function of problems for equations of elliptic and parabolic types.

A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problem with the initial condition being the delta function concentrated at a single plane (i.e. the plane wave). A certain associated operator is written in finite differences with respect to two out of three spatial variables, i.e. "partial finite differences". The grid step size is bounded from below by a fixed number. A Carleman estimate is applied to obtain, for the first time, a Hölder stability estimate for this problem. Another new result is an estimate from below the amplitude of the first term of the expansion of the solution of the forward problem near the characteristic wedge. [ABSTRACT FROM AUTHOR]

We present a loosely coupled, non-iterative time-splitting scheme based on Robin–Robin coupling conditions. We apply a novel unified analysis for this scheme applied to both a parabolic/parabolic coupled system and a parabolic/hyperbolic coupled system. We show for both systems that the scheme is stable, and the error converges as O (Δ t T + log (1 Δ t)) , where Δt is the time step. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior (as t → ∞ ) of solutions as an initial boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis ( x > 0 ). The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum ( σ (H 0) ) of the one-dimensional Schrödinger operator H 0 with periodic coefficients p (x) and q (x) . [ABSTRACT FROM AUTHOR]

In the half-space, three-parameter families of solutions are constructed for hyperbolic differential-difference equations with general-type translation operators acting with respect to all spatial variables. We prove that the obtained solutions are classical provided that the real part of the symbol of the corresponding differential-difference operator is positive. Classes of equations such that this condition is satisfied for them are provided. [ABSTRACT FROM AUTHOR]

Abstract In this paper, in a cylindrical domain D = Ω × ( 0 , T ) $D=\Omega \times (0,T)$ with Ω ⊂ R n $\Omega \subset {R}^{n}$ , we consider a mixed Cauchy problem with a potential lateral boundary condition for the following noncharacteristic degenerated equation L u = u t t − k ( t ) Δ x u ( x , t ) = f ( x , t ) , $$\begin{aligned} Lu=u_{tt}-k(t)\Delta _{x}u(x,t)=f(x,t), \end{aligned}$$ where k ( t ) ≥ 0 $k(t)\geq 0$ . As in the case for strictly hyperbolic equations, we first establish that u ∈ W 2 1 ( D ) $u\in W_{2}^{1}(D)$ and u ∈ W 2 2 ( D ) $u\in W_{2}^{2}(D)$ under the assumptions ∥ f k ∥ L 2 ( Ω ) ( t ) < ∞ $\Vert \frac{f}{k} \Vert _{L_{2}(\Omega )}(t)

In this paper, a nonlinear inverse boundary value problem for the second-order hyperbolic equation with nonlocal conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence (in a certain sense) to the original problem. Then using the Fourier method and contraction mappings principle, the existence and uniqueness theorem for auxiliary problem is proved. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the considered inverse coefficient problem is proved for the smaller value of time. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem in a strip for a two-dimensional hyperbolic equation containing the sum of a differential operator and a shift operator acting on a spatial variable that varies over the real axis. An operating scheme is used to construct the solutions of the equation. The solution of the problem is obtained in the form of a convolution of the function found using the operating scheme and the function from the initial conditions of the problem. It is proved that classical solutions of the considered initial problem exist if the real part of the symbol of the differential-difference operator in the equation is positive. [ABSTRACT FROM AUTHOR]

This paper is devoted to a sharp internal/boundary observability inequality for a hyperbolic system with a zero order potential. For this purpose, we first establish a new Carleman estimate for hyperbolic operator in H1-norm. Based on this Carleman estimate and a modified auxiliary optimal control problem, we obtain Carleman estimate for hyperbolic operator in L2-norm. Then, by virtue of a modified energy estimate and a delicate treatment of the observation region, we obtain an internal observability estimate with the observability constant of the order exp (C‖q‖2/3L∞(Q;ℝN×N)), with q the potential involved in the system. We also address the same problem for boundary observation. Compared with the related results in the literature, the main contributions of this paper are the observability constant is sharper, the waiting time T is shorter and the internal (or boundary) observation domain is smaller. [ABSTRACT FROM AUTHOR]

We study the existence of smooth solutions of a multidimensional hyperbolic equation containing the sum of differential operators and shift operators along arbitrary spatial coordinate directions. For this equation, we construct a three-parameter family of solutions. It is proved that the resulting solutions are classical under the condition that the real part of the symbol of the differential–difference operator in the equation is positive. Classes of equations for which this condition holds are given. [ABSTRACT FROM AUTHOR]

In this paper, we prove a stability result for a nonlinear wave equation, defined in a bounded domain of R N , N ≥ 2 , with time-dependent coefficients. The smooth boundary of Ω is Γ = Γ 0 ∪ Γ 1 such that Σ = Γ ¯ 0 ∩ Γ ¯ 1 ≠ ∅ . On Γ 0 we consider the homogeneous Dirichlet boundary condition and on Γ 1 we consider the Neumann boundary condition with damping term. The presence of time-dependent coefficients and, moreover, of the singularities generated by the condition Σ ≠ ∅ brings some technical difficulties. The tools are the combination of appropriate functional with the techniques due to Bey, Loheac, and Moussaoui [2] and new technical arguments. [ABSTRACT FROM AUTHOR]

The paper considers the inverse problem of determining the unknown coefficient on the righthand side of the hyperbolic equation. An additional condition for finding the unknown coefficient, which depends on the variable time, is given in the integral form. Theorems on the uniqueness, stability, and existence of the solution have been proved. [ABSTRACT FROM AUTHOR]

M.M. Lavrentiev's linear integral equation arises as a result of a special transformation of a nonlinear coefficient inverse wave sensing problem. The completeness of the set of products of regular harmonic functions and Newtonian potentials supported by a segment is proved. As a corollary, we establish the uniqueness of the solution to M.M. Lavrentiev's equation and a related inverse problem of wave sensing. We present results of an approximate solution of this equation by using parallel calculations. [ABSTRACT FROM AUTHOR]