Vol 28, No 3 (2017)

We consider the inverse problem for the two-dimensional modified FitzHugh–Nagumo model in the presence of an infarct. The inverse problem determines the myocardium excitation source function (a function of space variables and time) from a system of partial differential equations. Additional dynamic measurements of the potential are carried out on the entire inside boundary of the region representing a cross-section of the heart and its ventricles by a horizontal plane, which fits the real heart geometry. A numerical method is proposed for the solution of this inverse problem with the discrepancy functional; numerical results are reported.

We consider the dependence of the MHD stability of an electrolysis bath on the shape of the work space. As the optimal work-space shape we choose the one that achieves the best separation of the eigenvalues in the spectrum of the multidimensional problem posed for the kinematic equation for the electrolytealuminum interface in a particular electrolysis bath.

The article focuses on the analysis of photoplethysmograms. The quality of a photoplethysmogram is assessed by solving the classification problem, i.e., identifying the patient from the photoplethysmogram. Efficient patient identification methods are proposed, including methods based on DTW and TWED metrics. Identification accuracy reaches 70%.

We propose and investigate three mathematical models describing production cycles. They incorporate various mechanisms of endogenous fluctuations in economic systems. The models are based on ODE systems. The first model is a Keynesian IS-LM model of business cycles. The interest rate is determined by the money market and influences the relationship between savings and investments, allowing funds to flow from one to the other and vice versa. In the second case, the fluctuation mechanism is associated with time lags between investment growth, capital growth, and rate of return on capital. As a result, the economy periodically “overheats”, as rapid growth of capital suppresses the return rates, production becomes unprofitable, and investments sharply decline. Two models realizing this mechanism are proposed. One is a minimalist model based on a system of three ODEs. The other is an augmented model that sufficiently fully describes modern economic systems of developed countries and consists of nine ODEs and nine algebraic equations. It encompasses all the principal markets: labor market, capital market, financial market, and commodity market. Bifurcation analysis of the three models is carried out, oscillation regions are determined, and oscillation mechanisms are examined in detail. The model parameters are chosen so that the cycle periods are 12–17 years long.

Fractal interpolation is more general than the classical piecewise interpolation due to the presence of the scaling factors that describe smooth or non-smooth shape of a fractal curve/surface. We develop the rational cubic fractal interpolation surfaces (FISs) by using the blending functions and rational cubic fractal interpolation functions (FIFs) with two shape parameters in each sub-interval along the grid lines of the interpolation domain. The properties of blending functions and C¹-smoothness of rational cubic FIFs render C¹-smoothness to our rational cubic FISs. We study the stability aspects of the rational cubic FIS with respect to its independent variables, dependent variable, and first order partial derivatives at the grids. The scaling factors and shape parameters seeded in the rational cubic FIFs are constrained so that these rational cubic FIFs are convex/concave whenever the univariate data sets along the grid lines are convex/concave. For these constrained scaling factors and shape parameters, our rational cubic FIS is convex/concave to given convex/concave surface data.

We consider a modified Merton’s model of optimal consumption that allows for the utility of continuous and terminal consumption. An explicit solution of the Hamilton-Jacoby-Bellman equation is found. An upper bound is constructed on the probability of an event involving either investor ruin or negative consumption.