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We study nonlinear inverse problems for first order hyperbolic equations. We find a solution and an unknown coefficient depending on the time variable. For additional conditions we consider integral and boundary overdetermination conditions. We establish the existence of a regular solution, i.e., the solution possesses all generalized derivatives in the sense of Sobolev entering the equation. [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]

We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term. The coefficients of the equation are smooth functions constant beyond some compact domain in the three-dimensional space. A plane wave with direction falls to the heterogeneity from the exterior of this domain. A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for a time interval including the moment of arrival of the wave at these points. The unit vector is assumed to be a parameter of the problem and can run through all possible values sequentially. We study the inverse problem of determining the coefficient of the nonlinearity on using this information about solutions. We describe the structure of a solution to the direct problem and demonstrate that the inverse problem reduces to an integral geometry problem. The latter problem consists of constructing the desired function on using given integrals of the product of this function and a weight function. The integrals are taken along the geodesic lines of the Riemannian metric associated with the leading part of the differential equation. We analyze this new problem and find some stability estimate for its solution, which yields a stability estimate for solutions to the inverse problem. [ABSTRACT FROM AUTHOR]

Integral identities that are fulfilled for measure of difference between the exact solution of a parabolic-hyperbolic equation and any functions from a corresponding energy class are proved. These identities make it possible to derive two-sided a posteriori estimates for approximate solution to the corresponding Cauchy problem. The left-hand side of such an estimate is a natural measure of deviation from the solution, and the right-hand side depends on the problem data and the approximate solution itself and, therefore, it can be explicitly calculated. These estimates can be used to control the accuracy of approximate solutions and to compare solutions to Cauchy problems with different initial conditions. These estimates also allow one to quantitatively assess the effects occurring due to inaccuracies in the initial data and in the coefficients of the equation. [ABSTRACT FROM AUTHOR]

The iterated Crank-Nicolson (ICN) method is a successful numerical algorithm for solving partial differential equations. When unequal weights are used in the ICN predictor-corrector process, the convergence rate is reduced to first order. In this paper, we propose two modified ICN algorithms that achieve second order convergence rate, based on two different ways of choosing the weights. The first approach employs geometrically averaged weights in two consecutive iterations, and the second one uses arithmetically averaged weights for two consecutive time steps. The stability and second order accuracy of our methods are verified using stability and truncation error analysis and numerically demonstrated by solving linear and semi-linear hyperbolic partial differential equations, Burgers' equation, and the Gross-Pitaevskii equation. [ABSTRACT FROM AUTHOR]

The formulations and methods of solution of the problems of diagnostics of thermoelasticity on a stationary boundary of a solid body are considered. A weak formulation of the boundary-value problem is presented for a nonlinear hyperbolic equation (a nonstationary wave equation with density, elasticity modulus, and thermal stress depending on temperature) with mixed boundary conditions. An approximate solution of this problem in the space W 2 1 has been obtained, and an example of solution is given. Based on the weak formulation of the boundary-value problem, problems of diagnostics have been formulated in the sense of the theory of function traces. According to the theorem about the uniqueness of the relationship between the function and its trace in the space W 2 1 , a conclusion has been drawn about the need to control the convergence of the calculated wave function to the experimental one only on boundaries with unstable conditions in solving the problem of diagnostics. [ABSTRACT FROM AUTHOR]

The model of Timoshenko's shell theory of shells was used to analyze the dynamic characteristics of conical shells of variable thickness on a Pasternak elastic bed under nonstationary loading. Based on the Hamilton–Ostrogradsky variational principle, the equations of motion of a conical shell of variable thickness on a Pasternak elastic bed were derived. This system of hyperbolic differential equations is solved by the finite difference method. The numerical algorithm for solving the obtained equations is based on applying the integral-interpolation method for constructing difference schemes in the spatial coordinate and an explicit finite difference scheme for integration in the time coordinate. The influence of geometric dimensions, taper angle, and elastic media on the natural frequencies and other dynamic characteristics of a conical shell of variable thickness under the action of a pulsed load is analyzed using specific examples. New mechanical effects are revealed. [ABSTRACT FROM AUTHOR]