Home – Home – Jornals – Numerical Analysis and Applications 2013 number 4

In this paper we investigate the precision of estimate of the expectation of solutions to stochastic differential equations with a random structure. The dependence of the precision of estimate on the size of the integration step of the generalized Euler method and on the volume of the simulated trajectories is shown. A strong loss of the precision of estimate at deterministic or random times of changing the SDE structure is shown on an example of a simple equation. This requires the use of supercomputers for the statistical modeling. The results of the numerical experiments carried out in the Siberian SuperСomputer Center are presented.

The construction of the Newton-Cotes formulas is based on approximating an integrand by the Lagrange polynomial. The error of such quadrature formulas can be serious for a function with a boundary-layer component. In this paper, an analogue to the Newton-Cotes rule with four nodes is constructed. The construction is based on using non-polynomial interpolation that is accurate for a boundary layer component. Estimates of the accuracy of the quadrature rule, uniform on gradients of the boundary layer component, are obtained. Numerical experiments have been performed.

The inverse problem is solved by an optimization method using the Laguerre functions. Numerical simulations are carried out for the one-dimensional Maxwell's equations in the wave and diffusion approximations. Spatial distributions of permittivity and conductivity of the medium are determined from a known solution at a certain point. The Laguerre harmonics function is minimized. The minimization is performed by the conjugate gradient method. Results of determining permittivity and conductivity are presented. The influence of shape and spectrum of a source of electromagnetic waves on the accuracy of solution of the inverse problem is investigated. The accuracies of the solutions with a broadband and a harmonic sources of electromagnetic waves are compared.

In the computational experiment, the influence of heat exchange through top and bottom of the gas-bearing reservoir on the dynamics of temperature and pressure fields in the process of real gas production from a single well is investigated. The experiment was carried out with a modified mathematical model of non-isothermal gas filtration, obtained from the energy and mass conservation laws and the Darcy law. The physical and caloric equations of state together with the Newton-Rihman law of heat exchange of a gas reservoir with surrounding enclosing rocks are used as closing relations. It is shown that the influence of the heat exchange with environment on the temperature field of a gas-bearing reservoir is localized in a narrow zone near its top and bottom, though the size of this zone increases with time.

The A(α)-stable numerical methods (ANM) for the number of steps k ≤ 7 for stiff initial value problems (IVPs) in ordinary differential equations (ODEs) are proposed. The discrete schemes proposed from their equivalent continuous schemes are obtained. The scaled time variable t in a continuous method, which determines the discrete coefficients of the discrete method is chosen in such a way as to ensure that the discrete scheme attains a high order and A(α)-stability. We select the value of α for which the schemes proposed are absolutely stable. The new algorithms are found to have a comparable accuracy with that of the backward differentiation formula (BDF) discussed in [12] which implements the Ode15s in the Matlab suite.

Spline approximation with a reproducing kernel of a semi-Hilbert space is studied. Conditions are formulated that uniquely identify the natural Hilbert space by a reproducing kernel, a trend of spline, and the approximation domain. The construction of spline with external drift is proposed. It allows one to approximate functions having areas of big gradients or first-kind breaks. The conditional positive definiteness of some known radial basis functions is proved.

An algorithm for solving the optimal nonlinear filtering problem by statistical modeling is proposed. It is based on reducing the filtration problem to the analysis of stochastic systems with terminating and branching paths, using a structure similarity of the Duncan-Mortensen—Zakai equations and the generalized Fokker-Planck-Kolmogorov equation. The solution of such problem of analysis can be approximately found by using numerical methods for solving stochastic differential equations and methods for modeling inhomogeneous Poisson flows.

The probability of the precession of a pendulum with the Cardano suspension in conditions of an oscillation point of suspension based on the mathematical proof is investigated.