Vol 18, No 3 (2014)

A nonlocal problem for mixed type equation with a singular coefficient and the spectral parameter is formulated in the field, which hyperbolic part is vertical half-strip and elliptic part is rectangle. The nonlocal condition of problem combines the values of required function on the right and left boundaries of half-stripe and rectangle. The only requirement on the unknown function in the change type line is continuity. To research the given problem we apply the spectral method. The uniqueness and existence of a solution are proved. The solution is constructed as biortogonal series. Coefficients of this series should require special ODE systems, solved in the paper. The uniform convergence of the series is proved with the restrictions on problem conditions.

Inverse problems on restoration of coefficients to the differential equations with partial derivatives are of interest in many applied researches. These problems lead to necessity of the approached decision of inverse problems for the equations of mathematical physics which are incorrect in classical sense. In the article the existence, uniqueness and stability of the solution of the given inversion problem for the elliptic equation are proved.

We consider Volterra equation with two-variable, commonly encountered in the theory of elasticity. The purpose is to find new variants of sufficient conditions for it’s solvability in explicit calculation. The reduction principle of the original equation, first, to Goursat problem for differential equation of third order, and after that to two problems solving consecutively for equations of the first and second order is devised. One of these problems can be solved by direct equation integration, and the other’s solution can be written through Riemann function for which variants of its explicit construction are found. Seven variants of conditions for mentioned calculation were obtained in terms of coefficients of the original equation. Considering that there are four variants of factorization of equation of third order involved into the reasoning, virtually there are 28 variants of conditions for original equation solvability in quadratures noted in this article.

The boundary value problem of steady-state creep for thick-walled misaligned tube under internal pressure was considered. The approximate analytical solution of this problem by method of small parameter including the second approach is under construction. The solution for the state of plane deformation is constructed. The hypothesis of incompressibility of material for creep strain is used. As a small parameter the misalignment of the centers of the inner and outer radii of the tube is used. The main attention to the convergence of the resulting analytical solution considering the second approximation and assessment of its error is paid. It is noted that the convergence problem is solved only for boundary value problems in the theory of elasticity. Therefore the error assessment in the problem is solved on the basis of a comparison of the approximate analytical solution with the numerical solution constructed on the finite element method, for some special cases. Considering the symmetry of the problem, the finite element model was built for the half tube. The number of finite elements is about 18,000. Considering the symmetry of the problem the second half of the tube is replaced by boundary conditions. Analysis of analytical and numerical solutions is executed depending on the steady-state creep nonlinearity parameter and misaligned parameter that is ratio of the misalignment of the centers of the outer and inner diameter to the outer radius. It is shown that the error of deviation of the approximate analytical solution in the second approximation from numerical solution until the misalignment value of the centers of the inner and outer diameters of 0.1 for the tubes with small exponent of the steady-state creep (3 to 8) is not more than 9 %, and error to 8 % for the tubes with a large exponent of the steady-state creep nonlinearity is observed in the misaligned parameter to 0.06. Results of computations are presented in tabular form and in the form of graphs. Recommendations for the use of the constructed approximate analytical solution in applied problems are given.

The deformation process of the medium with full strains equaled to the sum of elastic and creep strains is considered. Elastic strains are described by Hook’s law while creep strains velocities are functions of stress components and some structural parameters, velocities of their changing are described by Rabotnov’s kinetic equations. It is assumed that Drucker’s stability postulate in big, formulated for materials with time-depending behaviour, is valid for creep strains. The inversion of depends between stresses and strains as well as uniqueness of the solution of boundary value problems are discussed. A special case of aforementioned creep equations for a strengthening material, when the strengthening parameter is the value of specific scattered creep energy, is considered. The sufficient conditions for Drucker’s stability postulate fulfillment in big are determined for this case, the reasons in favor of necessity of these conditions are given.

This article is devoted to a new method of ill-conditioned linear algebraic systems solving with the help of differentiation operator. These problems appear while solving the first kind integral Fredholm equations. The most difficult thing about this method is that differential operator discrete analogue matrix is rank deficiency matrix. The generalized singular value decomposition methods are used to solve those problems. The approach has high computational complexity. This also leads to additional computational error. Our method is based on the original regularized problem transformation into equivalent augmented regularized normal equation system using differential operator discrete analogue. The problem of spectrum matrix investigation of augmented regularized normal equation system with rank deficiency differential operator discrete analogue matrix is very relevant nowadays. Accurate eigenvalue spectrum research for this problem is impossible. That is why we estimated spectrum matrix bounds. Our estimation is based on a wellknown Courant-Fisher theorem. It is shown that estimated spectrum matrix bounds are rather accurate. The comparison between the proposed method and standard method based on the solving of normal system of equations is done. As shown in the paper, the condition number of normal method matrix is bigger than the condition number of augmented normal equations method matrix. In conclusion test problems description is given which proves our theoretical background.

The given paper contains the proof of that the number of combinatorial arrangements coincides with free components of the sums of equal powers with the natural bases and parameters in the presence of the simple equality connecting elements of these arrangements. In the proof the modified exposition of the components participating in formation of the sum of equal powers is used. This exposition becomes simpler and led to an aspect of product of binomial factors. Other variants of construction of corresponding product of binomial factors do not exist here. The received proof allows both to represent number of arrangements in the form of product, and to apply at this representation summation elements. Thus, the number of arrangements supposes characteristic expression not only in the form of product of its elements.

The article proposes a solution to the problem of mapping an algorithm from the field of Computational Mathematics on the target computing environment. The solution is based on a formal method for constructing parallel skeletons. The method comprises a specification of concurrency with the directed graphs and a formula for interpretation of dynamic behavior of such graphs. This interpretation is based on Temporal Logic of Actions approach proposed by Leslie Lamport. To illustrate the use of the method the “bag-oftasks” parallel skeleton is discussed hereinafter. We present graphically basic skeleton operations with the proposed computational model. After that we specify a learning algorithm of hyper-radial basis function neural network in the terms of skeleton operations as a case study. This made it possible to parallelize the leaning algorithm and map it on desired computing environments with predefined run-time libraries. Computational experiments confirming that our approach does not reduce the performance of the resulting programs are presented. The approach is suitable for researchers not familiar with parallel computing. It helps to get a reliable and effective supercomputer application both for SMP and distributed architectures.