# No 3 (2016)

## Articles

### On an Auxiliary Nonlinear Boundary Value Problem in the Ginzburg-Landau Theory of Superconductivity and its Multiple Solutions

#### Abstract

We realize analytic-numerical investigation of a homogeneous boundary value problem (BVP) for a second-order ordinary differential equation (ODE) with cubic nonlinearity and two real parameters which arises from the Ginzburg-Landau theory of superconductivity. Multiple nontrivial solutions to this problem depending on the specified parameters are expressed through the Jacobi elliptic functions and describe the stationary states (near the critical values of temperature) of a superconducting infinite plate of a finite thickness without magnetic field. It is a “degenerate” problem with respect to the original nonlinear BVP for a superconducting plate in a magnetic field and is important to construct algorithm for finding all the solutions to the indicated input problem in a wide range of the parameters. Studied problem is of separate mathematical interest by itself.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):5-20

5-20

### Aggregated Dynamic Model of a Two-Sector Economy with Venture Investment

#### Abstract

Venture capital now is a significant part of investments in innovative projects. In this article it is proposed dynamic model of a two-sector economy with venture investment. There are five economic agents: the population, the banks, resellers and producers, divided into two sectors the traditional and innovative. The article gives microdescription of firms in both sectors. The main sector companies at any given time are founded by moving into it one of the venture sector firms, where several firms are being created in each moment. Innovative firms parameters are set by the normal distribution. Venture investor takes loans of the banking system, and is fully in control of gains and losses of the venture sector. People invest money in the firms which join the traditional sector to gain their share in profits. The article describes the process of exiting the innovation sector by firms and their sale. Also it is described the process of liquidation of unprofitable firms in both sectors. The results of numerical experiments with a closed mathematical model are presented. The proposed model shows the exponential growth in which stands a distinctive kink in the time when the investments of the population reaching the level of bank loans.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):21-30

21-30

### Solution of the Boundary-Value Problem for a Systems of ODEs of Large Dimension: Benchmark Calculations in the Framework of Kantorovich Method

#### Abstract

We present benchmark calculations of the boundary-value problem (BVP) for a systems of second order ODEs of large dimension with help of KANTBP program using a finite element method. In practice, for solving the BVPs with the long-range potentials and a large number of open channels there is a necessity of solving boundary value problems of the large-scale systems of differential equations that require further investigation of convergence and stability of the algorithms and programs. With this aim we solve here the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the parametric basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated in an analytical form as solutions of the auxiliary parametric Sturm-Lioville problem for a second-order ODE. As a result, the two-dimensional problem is reduced to a boundary-value problem for a set of self-adjoint second-order ODEs for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method. The efficiency, stability and convergence of the calculation scheme is shown by benchmark calculations for a triangle membrane with a degenerate spectrum.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):31-37

31-37

### Algorithms and Programs for Solving Boundary-Value Problems for Systems of Second-Order ODEs with Piecewise Constant Potentials: Multichannel Scattering and Eigenvalue Problems

#### Abstract

The new algorithms and programs, implemented in Maple, for solving waveguide-type multichannel scattering and eigenvalue problems for systems of the second-order ODEs with N х N matrix piecewise constant coefficients on the axis are proposed. New algorithm and program for solving the boundary-value problems by method of matching the fundamental solutions (MMFS) of the system of ODEs at the points of discontinuity of potentials are elaborated. In each of the subintervals of an axis the general solution of the system of ODEs are sought in the form of linear combination of 2N fundamental solutions with unknown coefficients. Each fundamental solution explicitly dependent on spectral parameter and eigenvalues and eigenvectors of algebraic eigenvalue problems with N х N matrix of constant potentials. From the condition of continuity for the solutions and their derivatives at the discontinuity points of the potentials, the system of algebraic equations is followed. In the case of bound or metastable state problem the obtained system of algebraic equations contains nonlinear dependence of unknown spectral parameter. For solving such nonlinear problem symbolic-numerical algorithm is formulated. The benchmark calculations of bound, metastable and scattering states of BVPs for systems of the second-order ODEs obtained using program of the MMFS are compared with those obtained using program of the finite element method.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):38-52

38-52

### Integral Model of Natural Convection Turbulent Boundary Layers Next to Heated Vertical Surface width a Large Lateral Grasgof Number and Homogeneous Heat Flux

#### Abstract

The proposed integrated two-zone model to describe the characteristics of the turbulent free convection boundary layer near a vertical wall. To obtain accurate profiles of vertical velocity and excess temperature, taking into account the effect of wall region of the flow in the main part of the boundary layer. Offered the correct way of using the Blasius formula to determine the value of turbulent shear stress on the wall. On the basis of the differentiation profile of excess temperature the relation connecting the specific heat flux and excess wall temperature. It is shown that in the framework of the chosen approximation ratio linking density of the heat flux and excess wall temperature has a form similar to the formulas Vliet-Ross and Saunders. The obtained closed system of integro-differential equations describing the flow in free convection flow the boundary layer. In the framework of the chosen approximation, the total system of integro-differential equations was reduced to a system of nonlinear ordinary differential equations of the first order. On the basis of the obtained systems was carried out numerical simulation of a natural convection turbulent boundary layer in terms of the number of experiments. Made comparison of results of numerical simulation, including the fields of vertical velocity and excess temperature, with experimental data.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):53-60

53-60

### Analysis of the Flow of the Near Wall Region in Natural Convection Boundary Layer

#### Abstract

The analysis of the flow of turbulent natural convection boundary layer near a heated vertical plate was done. On the basis of comparison of criteria of similarity determined relative influence of viscous and convective forces in this region. The approximate equations describing the flow characteristics in the near wall region takes into account the relative influence of viscous and convective forces. Using the analogy between equations of forced turbulent boundary layer and obtained the equations for the near wall region was found corresponding profiles of vertical velocity and excess temperature. On the basis of the profiles of vertical velocity and excess temperature were built of the velocity field and temperature in the near wall region. In the fields of velocities obtained an expression describing the friction in turbulent wall region of a turbulent boundary layer free convection flow. Based on the analogy with a forced turbulent boundary layer and the flow in the near wall region of natural convection turbulent boundary layer near a vertical plate has been proposed to use the Blasius formula for finding the values of the turbulent shear stress on the wall. A review of the results was done.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):61-65

61-65

### The Peculiarities of Acceleration of Ultra-Relativistic Electrons by the Powerful Laser Radiation at the Regime of Cyclotron Autoresonance

#### Abstract

Analysis of the numerical solution of the motion equations of electrons in the field of ultrashort laser pulse, propagating along the steady magnetic field at the conditions of cyclotron autoresonance, is performed. It is shown that in the absence of the condition of cyclotron resonance during injection of electrons they momentarily escape the range of interaction. Laser radiation of the circular polarization is defined in the paraxial approximation in the form of the Gaussian beams of the basic and the first modes. Corrections of the first approximation to the components of the radiation field are taken into consideration. Calculations show that corrections of the first order to the transverse components exert no sufficient influence on the autoresonant motion of electrons whereas the longitudinal components of the first approximation play a major role. It is shown also that the specific form of the pulse is inessential. Images of the spatial distribution of the vectors of the radiation field in the transverse plane depending on the longitudinal coordinate (the direction of the radiation propagation) are obtained. It is shown that the character of changes of energy of an electron beam essentially depends on their position of injection in the focal plane. In this case acceleration as well as deceleration is possible in dependence on the position of injection of electron beam. It is shown that under the optimal conditions of injection the mechanism of the cyclotron autoresonance can provide sufficiently high efficient of acceleration of ultrarelativistic electrons in the field of powerful laser radiation with sufficiently high average rate at the distance of the order of two Rayleigh lengths. The basic mode is more preferable due to more simple description of that mode, higher acceleration rate and wider acceleration zone of injection of an electron beam.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):66-78

66-78

### Numerical Simulation of a Short Ion Beam Interaction with Plasma

#### Abstract

The problem of a short ion beam passing through the plasma layer is considered in this paper. In such a process the plasma electrons capture by the beam takes place. The charge compensation is necessary during the charged beams transport with the purpose to prevent their dispersion under the influence of the own Coulomb field. It is important to investigate the methods of the beam charge compensation methods for the reason of their numerous applications. Specifically the active investigations of the intensive ion beams interactions with the thermonuclear targets in the controlled fusion problem are performed last years. In this paper the one-dimensional electrostatic approximation is used and the conditions of its applicability are presented. The electron movement in the ion beam field with the model density distribution is considered. It is shown by the numerical simulation using the particle-in-cell method that during the short ion beam passing through the plasma layer the capture of the part of plasma electrons by the beam field takes place. But unlike the hydrodynamical description used by other authors this process has the essentially kinetic nature moreover the collective electric field is compared with the beam field. The beams of accelerated electrons are formed under the influence of the total field leading to the nonlinear regime of the beam instability and strong heating of the plasma electrons. It is shown that the oscillating field caused by the plasma oscillations on the plasma boundaries affects essentially on the electron capture. The numerical simulation of the beam passing through the plasma layer on the time intervals compared with ion plasma period is carried out. The particle-in-cell method is applied in this case for the ion movement calculation. It was supposed that the electrons have the Boltzmann density distribution in the self-consisted field. The boundary problem for the Poisson equation which becomes nonlinear in such a statement was solved numerically by the shooting method. It was demonstrated the formation of the stationary structures of the ion phase space holes type for the electron temperature much larger the ion one.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):79-86

79-86

### Information about the authors

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):87-88

87-88

### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2016;(3):89-90

89-90