Vol 29, No 1 (2018)

Numerical methods are proposed for determining the initial condition in Cauchy problems for a hyperbolic equation with a small parameter multiplying the highest-order derivative. Additional information for the inverse problem is provided by the solution of the Cauchy problem specified at x = 0 as a function of time. Results of numerical calculations illustrating the potential of the proposed method are reported.

Using grids (meshes) formed from polyhedra (polygons in the two-dimensional case), we consider differential and boundary grid operators that are consistent in the sense of satisfying the grid analog of the integral identity – a corollary of the formula for the divergence of the product or a scalar by a vector. These operators are constructed and applied in the Mimetic Finite Difference (MFD) method, in which grid scalars are defined inside the grid cells and grid vectors are defined by their local normal coordinates on the planar faces of the grid cells. We show that the basic grid summation identity is a limit of an integral identity written for piecewise-smooth approximations of the grid functions. We also show that the MFD formula for the reconstruction of a grid vector field is obtained by approximation analysis of the summation identity. Grid embedding theorems are proved, analogous to well-known finite-difference embedding theorems that are used in finite-difference scheme theory to derive prior bounds for convergence analysis of the solutions of finite-difference nonhomogeneous boundary-value problems.

The article analyzes computations of low-frequency electrodynamic fields in a nonhomogeneous conducting medium using the integral equation method. Contrast effects arise when the nonhomogeneities are embedded in a low-conductivity medium. As the conductivity of the enclosing medium decreases, tighter grid spacing is required in the nonhomogeneous region.

The problem of approximation of electrocardiograms has a great practical importance and it is constantly in the field of research. Solution of this problem allows us to automate and computerize the process of diagnosis and to detect timely deviations from normal characteristics of cardiograms of heartbeat in the early stages of appearance and development of a disease. There are different methods for approximation of cardiograms. All these methods have their drawbacks. The article deals with new methods of approximation of cardiograms, which do not have any disadvantages of the known methods. The developed mathematical models are based on new methods of approximation of step functions, and these methods can significantly reduce errors of the approximation of cardiograms. In this paper new methods of approximation of step functions with estimation of errors of the approximation are supposed. The proposed methods do not have any drawbacks of traditional expansions of step functions in Fourier series and can be used in problems of mathematical modeling of a wide class of processes and systems, in particular, for solution of medical and biological problems.

The article presents numerical modeling of the formation of a solitary wave in an annular channel under the action of a continuous wind generated by several sources distributed across the channel. The numerical results are compared with experimental data. Efficient application of parallel computational resources, produces a good result in acceptable time. The proposed model simulates the formation of solitary waves and the interaction of two nonlinear waves, and also produces the conditions of formation of 1, 2, 3, and 4 waves on a liquid surface.

We investigate the optimal management of a portfolio consisting of a risk asset and a risk-free bond. The asset dynamics is specified by the M-CEV model. Assuming that the investor has a power utility function, we derive an explicit analytical portfolio-management formula that contains confluent hypergeometric functions. The asymptotic form of the proposed portfolio management strategy is obtained, as well as approximate formulas consisting only of elementary functions. We additionally present an application of our results in the context of algorithmic statistical-arbitrage strategies.

We consider a nonlinear ordinary differential equation associated with a number of inverse scattering problems in acoustic and seismic sounding in which acoustic impedance and an impedance-dependent unknown damping coefficient are the unknown function. We prove that the Cauchy problem is uniquely solvable when the derivative is treated as a generalized function. It is established that the inverse scattering problem in a layered dissipative medium simultaneously determines the acoustic impedance and the damping coefficient. A regularized numerical algorithm is proposed and numerical results are reported.