Vol 57, No 10 (2017)

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.

First- and second-order numerical methods for optimizing controlled dynamical systems with parameters are discussed. In unconstrained-parameter problems, the control parameters are optimized by applying the conjugate gradient method. A more accurate numerical solution in these problems is produced by Newton’s method based on a second-order functional increment formula. Next, a general optimal control problem with state constraints and parameters involved on the righthand sides of the controlled system and in the initial conditions is considered. This complicated problem is reduced to a mathematical programming one, followed by the search for optimal parameter values and control functions by applying a multimethod algorithm. The performance of the proposed technique is demonstrated by solving application problems.

A piecewise interpolation approximation of the solution to the Cauchy problem for ordinary differential equations (ODEs) is constructed on a set of nonoverlapping subintervals that cover the interval on which the solution is sought. On each interval, the function on the right-hand side is approximated by a Newton interpolation polynomial represented by an algebraic polynomial with numerical coefficients. The antiderivative of this polynomial is used to approximate the solution, which is then refined by analogy with the Picard successive approximations. Variations of the degree of the polynomials, the number of intervals in the covering set, and the number of iteration steps provide a relatively high accuracy of solving nonstiff and stiff problems. The resulting approximation is continuous, continuously differentiable, and uniformly converges to the solution as the number of intervals in the covering set increases. The derivative of the solution is also uniformly approximated. The convergence rate and the computational complexity are estimated, and numerical experiments are described. The proposed method is extended for the two-point Cauchy problem with given exact values at the endpoints of the interval.

An X-ray tomography problem that is an inverse problem for the transport differential equation is set up and investigated. The absorption and single scattering of particles are taken into account. The transport equation is nonstationary (its coefficients and the unknown function depend on time), involves multiple energy levels, and its coefficients can undergo jump discontinuities with respect to the spatial variable (in other words, the medium in which the process proceeds is inhomogeneous). The sought object is the set on which the coefficients of the equation suffer a discontinuity, which corresponds to the search for the boundaries between the different substances composing the sensed medium.

The incompressible flow over a stationary periodically irregular surface and a compact body of revolution is computed by applying a modified discrete source method. The method is tested as applied to the flow over a stationary sphere. Numerical results are presented for flows over a sinusoidal surface, a cycloidal surface, a spheroid, a Chebyshev particle, an “analytical” cylinder, and a half ball.

The onset of convection in a porous anisotropic rectangle occupied by a heat-conducting fluid heated from below is analyzed on the basis of the Darcy–Boussinesq model. It is shown that there are combinations of control parameters for which the system has a nontrivial cosymmetry and a one-parameter family of stationary convective regimes branches off from the mechanical equilibrium. For the two-dimensional convection equations in a porous medium, finite-difference approximations preserving the cosymmetry of the original system are developed. Numerical results are presented that demonstrate the formation of a family of convective regimes and its disappearance when the approximations do not inherit the cosymmetry property.