Computational Mathematics and Modeling

We consider a problem that describes the biorthogonal adjoint system for the classical system of root functions for a loaded string with one free end in the presence of a multiple eigenvalue. We show that the complete and minimal subsystems isolated from the system of root functions constitute a basis in the presence of one or two associated functions.

We investigate the general boundary-value problem for the operator lu = −u′′ + q(x)u , 0 < x < 1, If the complex-valued coefficients q(x) is summable on (0,1), the integral \( {\int}_0^1x\left(1-x\right)\left|q(x)\right| dx \) converges.

The asymptotic solutions of the equation lu = μ²u derived in this article are used to construct the asymptotic spectrum of the problem, to classify the boundary conditions, and to prove theorems asserting that the system of root functions is complete and forms an unconditional basis in L₂ (0,1).

A mathematical model is developed for the blood flow in the vessels connected with the hepatic portal vein. A satisfactory anatomical model of the venous vessels of unpaired organs in the abdominal cavity and in the portal vein basin is constructed and integrated into the general systemic circulation model. Computer experiments are carried out modeling redistribution of venous and arterial blood flows in the presence of portal hypertension in liver fibrosis. The hydrodynamic properties of the blood flow are investigated allowing for anatomical and artificial shuts and their effect on pressure reduction in the portal vein. The calculation results are consistent with clinical data.

A mathematical model is constructed for the first-wall cooling system in the tokamak reactor. A numerical analysis is carried out for the existing first-wall design for the future FNS (Fusion Neutron Source) reactor.

This paper is devoted to an analysis of the rate of deep belief learning by multilayer neural networks. In designing neural networks, many authors have applied the mean field approximation (MFA) to establish that the state of neurons in hidden layers is active. To study the convergence of the MFAs, we transform the original problem to a minimization one. The object of investigation is the Barzilai–Borwein method for solving the obtained optimization problem. The essence of the two-point step size gradient method is its variable steplength. The appropriate steplength depends on the objective functional. Original steplengths are obtained and compared with the classical steplength. Sufficient conditions for existence and uniqueness of the weak solution are established. A rigorous proof of the convergence theorem is presented. Various tests with different kinds of weight matrices are discussed.

We consider a mathematical model of the treatment of psoriasis on a finite time interval. The model consists of three nonlinear differential equations describing the interrelationships between the concentrations of T-lymphocytes, keratinocytes, and dendritic cells. The model incorporates two bounded timedependent control functions, one describing the suppression of the interaction between T-lymphocytes and keratinocytes and the other the suppression of the interaction between T-lymphocytes and dendritic cells by medication. For this model, we minimize the weighted sum of the total keratinocyte concentration and the total cost of treatment. This weighted sum is expressed as an integral over the sum of the squared controls. Pontryagin’s maximum principle is applied to find the properties of the optimal controls in this problem. The specific controls are determined for various parameter values in the BOCOP-2.0.5 program environment. The numerical results are discussed.

Different forms of control problems for linear systems under essential uncertainty are considered. Specifically, three problems are examined. First, the unknown disturbances are assumed bounded and only their bounds are known. The problem is solved under various assumptions regarding the order of the disturbances and the observed signals, as well as the properties of the system. Second, the estimation of the unknown input (i.e., the inversion problem or the inverse problem) is considered. This problem is solved by the controlled model method with the control designed to stabilize the difference between the observed outputs of the original system and the model. Robust stabilization algorithms produce estimates of the unknown signals with a desired accuracy. The third problem focuses on stabilization of switched systems. The dynamics of the chosen system is described at each instant by one of the systems from a given finite set. Switching between regimes may depend both on time and on the system phase vector. Two problems are solved successively: finding stabilizers (a unique stabilizer if possible) for each plant from the family, and then investigating the conditions when switching between stable regimes does not disrupt the stability of the switched system. Various stabilization methods are obtained for switched systems covering different switching schemes.

We consider the maximization of the welfare function in the process of expansion of the energy transmission network. Production and transportation costs as well as the utility of consumption are taken into consideration. The article presents previously developed methods for the calculation of optimal (or nearly optimal) transmission capacities, flows, and production volumes with a model of a network energy market. We also examine the dynamic problem of optimal expansion of the transmission system given that the demand and cost functions are exogenous for each time interval during the planning period,