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.
A sample of values of plasma density in a thermonuclear facility is studied. A methodology for processing experimental data that makes it possible to establish correspondence between this sample and a model of nonstationary noise is proposed. This model is formed as convolution of a stationary sequence and a memory function, and it makes it possible to simulate the competition between space and time nonlocalities. A physical interpretation of the nonlocality parameters is described.
The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.
For a nonlinear controlled system, a fixed-time approach problem is considered in which the target point location becomes known only at the start of motion. According to the proposed solution method, node resolving program controls corresponding to a finite collection of target points from the set of their admissible locations are computed in advance and a refined control for the target point given at the start of motion is determined via linear interpolation of the node controls. The procedure for designing such a resolving control is formulated in the form of two algorithms, one of which is run before the start of the motion, and the other is executed in real time while the system is moving. The error in the transfer of the system’s state to the target point by applying these algorithms is estimated. As an example, we consider the approach problem for a modified Dubins car model and a target point about which only a compact set of its admissible locations is known before the start of motion.
A formal asymptotic expansion is constructed for the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearities and weak diffusion in the case of several space variables. Under the conditions imposed on the data of the problem, the leading asymptotic term is described by the multidimensional generalized Burgers–Korteweg–de Vries equation. Under certain conditions, the remainder is estimated with respect to the residual.
The paper presents the results of solving a boundary value problem for a system of two integro-differential equations that simulate the action of an external electric field on a plasma layer. This system is an implication of the Boltzmann–Maxwell equations, and the physical meaning of the sought functions is the strength of a self-consistent electric field and perturbation of the electron distribution density. The solution of the problem is constructed using the theories of Fourier transform of generalized functions and singular integral equations with the Cauchy kernel. The dependence of the solution on the frequency of the external field is studied.
In this work, we propose a numerical method for solving a singularly perturbed convection-diffusion problem that involves a time delay term. A priori bounds and properties of the continuous solution are discussed. Using the backward Euler method for the time derivative term, the problem is approximated by a set of singularly perturbed boundary value problems. Then, using a higher-order finite difference method, the boundary value problem is approximated on a piecewise uniform Shishkin mesh. The stability analysis of the method is studied using the comparison principle and discrete solution bounds. We proved that the proposed scheme is uniformly convergent, with an order of convergence of almost two in space and one in time. Two numerical examples are considered to validate the applicability of the proposed scheme. The proposed scheme has better accuracy than some schemes in the literature.
Unique solvability of systems of linear algebraic equations is studied to which many inverse problems of geophysics are reduced as a result of discretization after applying the method of integral equations or integral representations. Examples of singular and nonsingular systems of various dimensions that arise when processing magnetometric and gravimetric data from experimental observations are discussed. Conclusions are drawn about methods for constructing an optimal network of experimental observation points.
The problem of minimizing (maximizing) multiple extremum functionals (infinite-dimensional optimization) is considered. This problem cannot be solved by conventional gradient methods. New gradient methods with adaptive relaxation of steps in the vicinity of local extrema are proposed. The efficiency of the proposed methods is demonstrated by the example of optimizing the shape of a hydraulic gun nozzle with respect to the objective functional, which is the average force of the hydraulic pulse jet momentum acting on an obstacle. Two local maxima are found, the second of which is global; in the second maximum, the average force of the jet momentum is three times higher than in the first maximum. The corresponding nozzle shape is optimal. Conventional gradient methods have not found any maximum; i.e., they were unable to solve the problem.
This paper introduces a novel approach for analyzing efficient numerical solution of a class of singularly perturbed parabolic convection-diffusion PDEs having a finite jump discontinuity mostly in the source function at the interior of the spatial domain. These PDEs often appear in mathematical modeling of the semiconductor devices; and solutions of such problems usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. The prime objectives of this paper are to overcome the theoretical challenge related to the monotonocity of the finite difference operator which utilizes one-sided second-order difference operators at the interface point; and also to establish higher-order numerical approximation in space, regardless of smaller and larger values of the parameter ε. Proving discrete maximum principle of the proposed difference operator is found to be challenging task on the standard Shishkin mesh adapted to both boundary and weak interior layers. We overcome this difficulty by constructing a special non-uniform mesh, called modified layer-adapted mesh; and hereby establish the stability as well as the parameter-uniform error estimate in the discrete supremum norm. Finally, the theoretical estimate is verified with numerical results for test examples with and without the exact solution. Moreover, numerical results are obtained for the semilinear PDEs and also compared with the implicit upwind scheme to exhibit the efficiency of the proposed algorithm.
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