Volume 56, Nº 9 (2016)

A group of iteratively regularized methods of Gauss–Newton type for solving irregular nonlinear equations with smooth operators in a Hilbert space under the condition of normal solvability of the derivative of the operator at the solution is considered. A priori and a posteriori methods for termination of iterations are studied, and estimates of the accuracy of approximations obtained are found. It is shown that, in the case of a priori termination, the accuracy of the approximation is proportional to the error in the input data. Under certain additional conditions, the same estimate is established for a posterior termination from the residual principle. These results generalize known similar estimates for linear equations with a normally solvable operator.

To study the intensity of radiation transmitted through a layer of substance, a Monte Carlo algorithm is developed based on the expansion of the corresponding angular distribution density in terms of orthonormalized polynomials with a “Lambert” weight. The algorithm is optimized so as to simplify the computations as much as possible. Even a small effect of polarization and the deviation of the angular distribution from the Lambert one can be estimated rather accurately by applying the algorithm.

For the problem of parametric optimization of the coefficient and the right-hand side of the linear global electric circuit equation, formulas for the first partial derivatives of an integral cost functional with respect to control parameters are obtained.

A version of the facility location problem (the well-known p-median minimization problem) and its generalization—the problem of minimizing a supermodular set function—is studied. These problems are NP-hard, and they are approximately solved by a gradient algorithm that is a discrete analog of the steepest descent algorithm. A priori bounds on the worst-case behavior of the gradient algorithm for the problems under consideration are obtained. As a consequence, a bound on the performance guarantee of the gradient algorithm for the p-median minimization problem in terms of the production and transportation cost matrix is obtained.

Nonlinear Klein–Gordon equations with fractional power and logarithmic potentials and with a variation in the φ⁴ potential are found for which the existence of long-lived stable spherically symmetric solutions in the form of pulsons is numerically established. Their mean oscillation amplitude and the frequency of the fast oscillation mode do not vary in the course of the numerical simulation. It is shown that the stability of these pulsons is explained by the presence of a potential well.

Elastic wave propagation in a porous medium is numerically studied by applying the grid-characteristic method. On the basis of direct measurements of reflected and transmitted wave amplitudes, the reflection and decay coefficients are investigated as depending on the degree of porosity (percentage of the pore volume) and on the type of the filling substance (solid, liquid, or nothing). The reflection and decay coefficients are shown to be closely related to the porosity of the medium, which can be used in geological applications (estimation of porosity) and engineering applications (acoustic response attenuation).

From the Vlasov–Boltzmann kinetic equation for a collisional degenerate plasma, the electron distribution function is constructed in the quadratic approximation in the electric field strength. A formula for calculating the electric current is derived. It is shown that nonlinearity leads to the rise of a longitudinal electric current directed along the wave vector. The longitudinal current is orthogonal to the known transverse classical current obtained in the linear analysis. When the collision frequency tends to zero, all results obtained for a collisional plasma pass into the corresponding results for a collisionless plasma. The case of small wavenumbers is considered. It is shown that, when the collision frequency tends to zero, the expression for the current passes into the corresponding expression for the current in a collisionless plasma. Graphic analysis of the real and imaginary parts of the current density is performed. The dependence of the electromagnetic field oscillation frequency and electron–plasma-particle collision frequency on the wavenumber is studied.

This study models the magnetohydrodynamic (MHD) three-dimensional boundary layer flow of viscoelastic fluid. The flow is due to the exponentially stretching surface. The heat transfer analysis is performed through prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The thermal conductivity is taken temperature dependent. Series solutions of velocities and temperatures are constructed. Graphical results for PST and PHF cases are plotted and analyzed. Numerical values of skin-friction coefficients and Nusselt numbers are presented and discussed.