Abstract: In this paper, we consider a time-fractional inverse problem in which the nonlinear boundary conditions contain an unknown function. A finite difference scheme will be proposed to solve numerically the inverse problem. This inverse problem is generally ill-posed. For this reason, we will employ the mollification regularization method with the generalized cross-validation criterion to find a stable solution. The stability and convergence of numerical solutions are investigated. Finally, some numerical examples are presented to illustrate the validity and effectiveness of the proposed method. PubDate: 2019-11-14

Abstract: 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. PubDate: 2019-11-06

Abstract: 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. PubDate: 2019-11-01

Abstract: 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. PubDate: 2019-11-01

Abstract: The article examines the kinetic equations of irreversible coagulation with a source of monomers and a sink of particles that exceed the maximum allowed size. Time-periodic solutions are known for the class of Brownian kernels. In this study, we analyze the effect of the monomer source intensity on the period and the amplitude of the particle concentration oscillations over time. The numerical results suggest that as the source intensity is increased, the oscillation amplitude increases while the oscillation period decreases, so that no qualitative changes are observed in the solution structure. A change in source intensity does not produce scaling of the model time and model concentrations of the particles per unit volume of the medium. PubDate: 2019-11-01

Abstract: The inverse problem of the determination of the unknown coefficient in an integro-differential equation is considered. Existence and uniqueness theorems are proved for the inverse problem. A numerical method for the determination of the unknown coefficient is proposed and substantiated. Numerical results illustrating the convergence of the method are reported. PubDate: 2019-11-01

Abstract: The development of cervical cancer cells from normal cells is caused by the human papilloma virus (HPV), and the progression can be described using a population model of the cells and free virus. We develop a mathematical model consisting of five compartments to describe the interactions between the human papilloma virus and four classes of epithelial and basal cells (susceptible, infected, precancerous, and cancerous) of cervix. In our mathematical model, we consider that the disease transmission rate from precancerous to cancerous cells is governed by a response function f(P) according to the risk and our cell immunity power which is dependent on the antibody genes p53 and pRb. So we have considered f(P) as three types of functions linear, Holling type II, and Holling type III. We analyze the local stability of the equilibrium points of each of the types in a comparative way and investigate analytically and numerically the parameters that play an important role in the progression towards the cervical cancer. Furthermore, we have taken some control strategies on the Holling type III functional response based on two types of drugs to eradicate the infected and cancer cell populations. PubDate: 2019-11-01

Abstract: 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. PubDate: 2019-11-01

Abstract: 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, PubDate: 2019-11-01

Abstract: 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. PubDate: 2019-11-01

Abstract: 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. PubDate: 2019-11-01

Abstract: 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). PubDate: 2019-11-01

Abstract: We consider the sale of k securities in n trades, with not more than one security per trade. The sale results are assessed using the competitive ratio of the sum of k highest security prices to the total sale revenue. Lorenz constructed the solution of the game for 2k ≤ n. In this article, the solution is obtained in the general case both for the competitive ratio and for the Savage regret criterion. PubDate: 2019-08-19

Abstract: Many constructions of cubic splines are described in the literature. Most of the methods focus on cubic splines of defect 1, i.e., cubic splines that are continuous together with their first and second derivative. However, many applications do not require continuity of the second derivative. The Hermitian cubic spline is used for such problems. For the construction of a Hermitian spline we have to assume that both the values of the interpolant function and the values of its derivative on the grid are known. The derivative values are not always observable in practice, and they are accordingly replaced with difference derivatives, and so on. In the present article, we construct a C1 cubic spline so that its derivative has a minimum norm in L2 . The evaluation of the first derivative on a grid thus reduces to the minimization of the first-derivative norm over the sought values. PubDate: 2019-08-19

Abstract: In this study, turbulence of the flow field as well as the flow free surface in a U-shaped channel located along a side weir under supercritical flow conditions are simulated using the RNG k-ε turbulence model and VOF scheme. Comparison of the numerical and laboratory results shows that the numerical model simulates the free surface and characteristics of the flow field with acceptable accuracy. Then, the effects of the upstream Froude number of the side weir on the flow pattern of the main channel are investigated. For all Froude numbers, in the vicinity of the inner bank a free surface drop occurs at the beginning of the side weir upstream, and a surface jump is formed at the final quarter of the side weir length. Across the surface jump, the kinematic energy of water increases and the potential energy decreases. The dividing stream surface and the stagnation zone dimensions increase as the Froude number increases. PubDate: 2019-08-19

Abstract: Methods for the separation of a mixture of three-parameter lognormal distributions are investigated theoretically and empirically in the context of modeling message transmission delays in a computer cluster communication environment. Delay modeling based on mixtures of three-parameter lognormal distributions is proposed and parameter estimation methods for such models are investigated. Identifiability of the given family of distributions is proved, which is a necessary condition for the mixture separation problem to be well-posed. The proposed problem-oriented modifications of the standard algorithms are shown to be superior to the traditional methods. PubDate: 2019-08-19

Abstract: We consider a model of nonlinear interaction of femtosecond pulses with a Kerr nonlinear medium, allowing for first and second order dispersion, nonlinear response dispersion, and mixed time and space derivatives. The invariants are constructed by a transformation of the generalized nonlinear Schrodinger equation that involves changing to new functions and reduces the original equation to a form without the nonlinear response derivatives and the mixed derivatives. Appropriate conservation laws are established for the transformed equation. The invariants derived in this article lead to conservative difference schemes and allow control of computer simulation results. PubDate: 2019-08-19

Abstract: We examine a new method for the calculation of the traveling wave characteristic in a layered medium. The seismic impedance tensor method is improved by introducing the notion of a potential function and new iterative relationships are obtained for isotropic layered media. Comparison of the dispersion equation roots between the generalized reflection-transmission coefficient method (GR/TC) and the seismic impedance tensor method (SIT) has detected some roots that do not exist in SIT. To check the accuracy of the roots in both methods, we have compared the experimental results between the classical transition matrix method (Thomson-Haskell) and our new method (SIT) for the same layered medium model. The experimental results show that the roots obtained by the impedance method coincide with the classical transition matrix method. This also shows that some roots obtained from the dispersion equation in GR/TC cannot be accurately determined. PubDate: 2019-08-19

Abstract: A ridge detection algorithm is proposed for tracing blood vessels on images of the ocular fundus. Multiscale non-maximum suppression is applied to the image Laplacian. The multiscale algorithm exploits the pyramidal fine structure similarly to the SIFT method. Anisotropic diffusion is used in preprocessing, which makes it possible to boost the value of the convolution of the Laplacian with the Gaussian on ridge structures. The proposed algorithm has been tested on the ophthalmological image database DRIVE. The proposed preprocessing has substantially improved the ridge detection quality. PubDate: 2019-08-19

Abstract: The aim of this work is to investigate the possibilities of using a high-order harmonic basis for solving some magnetostatic problems. We consider known methods with our basis and the approach earlier elaborated by the authors. We present numerical results of their comparison when solving a linear problem on sequences of meshes with various parameters h and p. For a nonlinear problem with respect to two scalar potentials, it is shown that this model, in the suggested new weak formulation, keeps the property of monotonicity. From the results of this work it may be concluded that the harmonic basis gives more exact approximations on adaptive meshes for the considered magnetostatic problems in comparison with the usual approach. PubDate: 2019-08-19