Journal of Computational and Engineering Mathematics

ON NUMERICAL SOLUTION IN THE SPACE OF DIFFERENTIAL FORMS FOR ONE STOCHASTIC SOBOLEV-TYPE EQUATION WITH A RELATIVELY RADIAL OPERATOR

2020-12-21T13:04:36+05:00

The paper presents graphs of the trajectories of numerical solutions to the Showalter -- Sidorov problem for one stochastic version of the Ginzburg -- Landau equation in spaces of differential forms defined on a two-dimensional torus. We use the previously obtained transition from the deterministic version of the theory of Sobolev type equations to stochastic equations using the Nelson -- Glicklikh derivative. Since the equations are studied in the space of differential forms, the operators themselves are understood in a special form, in particular, instead of the Laplace operator, we take its generalization, the Laplace -- Beltrami operator. The graphs of computational experiments are given for different values of the parameters of the initial equation for the same trajectories of the stochastic process.

A COUPLED UNIFIED DIFFERENCE METHOD FOR NUMERICAL SOLUTION OF THE INITIAL BOUNDARY VALUE FOURTH ORDER PARABOLIC PROBLEMS

2020-12-21T13:04:36+05:00

In this article we proposed a coupled unified difference method for the numerical solution for the fourth order parabolic problems. To simplify the complexity of higher order differential term, we introduced an intermediate function and hence the higher order boundary value problem is transformed into an equivalent system of reduced order boundary value problems. We have discussed the convergence of the proposed method. Numerical experiments are performed to test and approve the efficiency, accuracy of the proposed method.

MODELING THE PROFITABILITY OF LOAN OPERATIONS CORPORATE CLIENTS

2020-12-21T13:04:36+05:00

The article is devoted to the optimization of the bank's operating activities on the basis of mathematical modeling of the assessment of the future profitability of corporate clients in the implementation of credit transactions in order to increase the efficiency of decisions taken in the implementation of transactions that carry credit risk. The modeling is based on the use of the forecasted rate of return on risk-weighted assets -- PRoRWA (predicted RoRWA) -- the predicted value of the RoRWA indicator for the client for the next "rolling year" taking into account the assessment of the profitability of a new potential credit transaction. The modeling process takes into account the assessment of the total profitability of the client for the bank, including income from cross-sales -- the additional income of the bank in the implementation of a particular transaction. The proposed model was tested at one of the divisions of a commercial bank. The simulation results are recommended for use in pricing in the process of setting the limit values of the marginality of transactions for corporate clients, in assessing the level of risks assumed by the bank in the implementation of new credit transactions, as well as in the analysis of alternative options for increasing the profitability of the bank's equity capital.

DICHOTOMIES OF SOLUTIONS TO THE STOCHASTIC GINZBURG -- LANDAU EQUATION ON A TORUS

2020-12-21T13:04:36+05:00

We consider a stochastic analogue of the Ginzburg -- Landau equation in spaces of differential forms defined on a two-dimensional smooth compact oriented manifold without boundary. When studying the stability of solutions, the Ginzburg -- Landau equation is considered as a special case of a stochastic linear Sobolev-type equation. All considerations are carried out in spaces of random $K$-variables and $K$-"noises" on the manifold. As a manifold, we consider a two-dimensional torus, which is a striking example of a smooth compact oriented manifold without boundary. Under certain conditions imposed on the coefficients of the equation, we prove the existence of stable and unstable invariant spaces and exponential dichotomies of solutions. We develop an algorithm to illustrate the results obtained. Since there exists a smooth diffeomorphism between a map and a manifold, we reduce the question of stability of solutions on a two-dimensional torus to the same question on one of its maps. The developed algorithm is implemented in the Maple software environment. The results of the work are presented in the form of graphs of stable and unstable solutions, which are obtained for various values of the parameters of the Ginzburg -- Landau equation.

OPTIMIZING FIRE-FIGHTING WATER SUPPLY SYSTEMS USING SPATIAL METRICS

2020-12-21T13:04:35+05:00

The current trend in the construction of new urban areas involves the coordination of a comprehensive preliminary planning of all systems and networks of power supply, water supply, drainage complexes, laying of communication networks and other communication networks of construction objects with the layout of residential buildings. In the paper there is an optimization of the number and type of distribution of hydrants of external fire-fighting water supply and pumping stations using various metrics that measure distance. The optimization of the dependence of the type of hydrant water supply networks on pumping stations is considered, considering the real practical tasks of providing fire-fighting water supply, taking into account the size, location and shape of modern buildings. The analysis of maps and standards revealed that the concept of distance implies only distance in a straight line that does not correspond to the modern layout of the development of subdistricts and significantly complicates the ability of firefighters to work. In the first part of the work, we present a mathematical model that optimizes both the number and the location of the set of hydrants that fully serve a given development area. The optimization algorithm uses a metric different from Euclidean distance. At the same time, it is assumed that the developed models are applicable for various types of development of subdistricts (line building, regular building, cluster housing, sporadic building, etc.). In the second part of the paper, we optimize the number of placement of pumping stations and the type of pipeline network connecting the fire-fighting water supply and pumping stations. To this end, we use the computer determination of the coordinates of the Torricelli~-- Steiner point implemented for an arbitrary set of consumer points and various spatial metrics.