Approximate computation of unsteady moving-boundary flows in a porous medium

The approximate method of integral relations is used to compute two problems of unsteady moving-boundary flow in a porous medium in the cases of linear parallel and plane radial motion. The problems are associated with the design and operation of underground gas storage facilities in horizontal and slightly sloping aquifers. It is assumed that the process is isothermal and the gas is ideal. Numerical results obtained in various approximations are described and analyzed. The results are presented as plots.     

Solution of a kinetic equation for diatomic gas with the use of differential scattering cross sections computed by the method of classical trajectories

A collision integral is constructed taking into account the rotational degrees of freedom of the gas molecules. Its truncation error is shown to be second order in the rotational velocity mesh size. In the solution of the kinetic equation, the resulting collision integral is directly computed using a projection method. Preliminarily, the differential scattering cross sections of nitrogen molecules are computed by applying the method of classical trajectories. The resulting cross section values are tabulated in multimillion data arrays. The one-dimensional problems of shock wave structure and heat transfer between two plates are computed as tests, and the results are compared with experimental data. The convergence of the results with decreasing rotational velocity mesh size is analyzed.     

Classroom note: Application of the Sumudu transform to discrete dynamic systems

The Sumudu transform is an integral transform introduced to solve differential equations and control engineering problems. The transform possesses many interesting properties that make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and dynamic/dynamic-differential systems. In this note, application of the method of Sumudu transform to the solution of discrete dynamic systems is described. Derivations find ready application in recurrence relations. Solutions are obtained to certain first degree, second degree and third degree dynamic systems.     

An approach to constructing models of multiphase elastic porous media

A novel approach to describing the behaviour of multiphase elastic porous media is proposed. The average values of the physical quantities needed to describe the motions of porous media are formulated using an integral relation. The validity of this relation is taken as the fundamental hypothesis. The integral definition of the average values enables integral relations to be devised for the average values from the integral laws of conservation of mass, momentum and energy and the increase in entropy. Along with the average values, the integral relations contain new variables that can be identified with generalized thermodynamic forces, which can be used to take into account the phase interaction in a porous medium. The integral relations are used to derive differential equations for the rate of entropy change and Gibbs relations for a porous medium as a basis for obtaining the constitutive relations. Relationships between the thermomechanical parameters of the model are established from the Gibbs relations under additional assumptions. The equation for the rate of entropy change can be used to establish relations between the generalized thermodynamic forces and fluxes. A complete system of differential equations in the defining parameters, which describes the motion of multiphase elastic porous media, is finally obtained.     

Parametrization of the Cauchy problem for systems of ordinary differential equations with limiting singular points

The method of solution continuation with respect to a parameter is used to solve an initial value problem for a system of ordinary differential equations with several limiting singular points. The solution is continued using an argument (called the best) measured along the integral curve of the problem. Additionally, a modified argument is introduced that is locally equivalent to the best one in the considered domain. Theoretical results are obtained concerning the conditioning of the Cauchy problem parametrized by the modified argument in a neighborhood of each point of its integral curve.     

A semi-analytic solution for multiple curved cracks emanating from circular hole using singular integral equation

This paper provides an elastic solution for an infinite plate containing multiple curved edge cracks emanating from a circular hole. A fundamental solution is suggested, which represents a particular solution for a concentrated dislocation in an infinite plate with the traction free hole. The generalized image method and the concept of the modified complex potentials are used in the derivation of the fundamental solution. After using the fundamental solution and placing the distributed dislocations at the prospective sites of cracks, a singular integral equation is formulated. The singular integral equation is solved by using the curve length method in conjunction with the semi-opening quadrature rule. By taking an additional point dislocation at the hole center, the number of the unknowns is equal to the number of the resulting algebraic equations. This is a particular advantage of the suggested method. Finally, several numerical examples are given to illustrate the efficiency of the method presented. Numerical examinations are carried out and sufficient accurate results have been found.     

污染物在非饱和带内运移的流固耦合数学模型及其渐近解

污染物在非饱和带中运移过程是多组分多相渗流问题.在考虑气相的存在对水相影响的前提下,基于流固耦合力学理论,建立了污染物在非饱和带内运移的流固耦合数学模型.对该强非线性数学模型采用摄动法及积分变换法进行拟解析求解,得出了解析表达式.对非饱和带内的孔隙压力分布、孔隙水流速以及污染物的浓度在耦合与非耦合气相条件下的分布规律进行解析计算.对该渐近解与Faust模型的计算结果进行了对比分析,结果表明:该模型解与Faust解基本吻合,且气相作用以及介质的变形对溶质的输运过程产生较大的影响,从而验证了解析表达式的正确性和实用性.这为定量化预报预测污染物在非饱和带中迁移转化和实验室确定压力-饱和度-渗透率三者之间的关系提供了可靠的理论依据.     

The Poisson and the Biharmonic Equations in the Interior of a Convex Polygon

A new method for analyzing linear elliptic partial differential equations in the interior of a convex polygon was developed in the late 1990s. This method does not rely on the classical approach of separation of variables and on the use of classical integral transforms and therefore is well suited for the investigation of the biharmonic equation. Here, we present a novel integral representation of the solution of the biharmonic equation in the interior of a convex polygon. This representation contains certain free parameters and therefore is more general than the one presented in [1]. For a given boundary value problem, by choosing these free parameters appropriately, one can obtain the simplest possible representation for the solution. This representation still involves certain unknown boundary values, thus for this formula to become effective it is necessary to characterize the unknown boundary values in terms of the given boundary conditions. This requires the investigation of certain relations refereed to as the global relations. A general approach for analyzing these relations is illustrated by solving several problems formulated in the interior of a semistrip. In addition, for completeness, similar results are presented for the Poisson equation by employing an integral representation for the Laplace equation which is more general than the one derived in the late 1990s.     

The Elementary Solution of Triple Integral Equations and The Solution of Triple Series Equations Involving Associated Legendre Polynomials and their Application

A method is developed for the formal solution of an important class of triple integral equations involving Bessel functions. The solution of the triple integral equations is reduced to two simultaneous Fredholm integral equations and the results obtained are simpler than those of other authors and also superior for the purposes of solution by iteration. In the same manner the formal solution of triple series equations involving associated Legendre polynomials is presented. The solution of the problem is reduced to that of solving a Fredholm integral equation of the first kind. Finally to illustrate the application of the results an electrostatic problem is discussed.     

Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions

A numerical technique based on the spectral method is presented for the solution of nonlinear Volterra-Fredholm-Hammerstein integral equations. This method is a combination of collocation method and radial basis functions (RBFs) with the differentiation process (DRBF), using zeros of the shifted Legendre polynomial as the collocation points. Different applications of RBFs are used for this purpose. The integral involved in the formulation of the problems are approximated based on Legendre-Gauss-Lobatto integration rule. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme.     

Influence of nonisothermal effects on gas production in northern regions

The influence of mathematical model parameters on the dynamics of pressure and temperature fields at nonisothermal gas filtration is investigated in a numerical experiment. A nonlinear system of partial differential equations obtained from the energy and mass conservation laws and the Darcy law are used to describe the process, and physical and caloric equations of state are used as closing relations. The boundary conditions correspond to a given pressure drop at the bottomhole. It is shown that the influence of the temperature field on such integral characteristics as cumulative gas production is most pronounced at moderate pressure drops. The size of the zone of possible hydrate formation in a gas reservoir is determined in a particular example.     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

Numerical simulation of the unsteady aerodynamic interaction of two plane airfoil cascades

A method for the calculation of unsteady aerodynamic interaction of two plane airfoil cascades that are in relative motion in a subsonic flow of ideal gas is developed. This interaction provides a two-dimensional approximation of the flow in a stage of an axial turbomachine. The method is based on the reduction of the problem to the calculation of the unsteady flow in a single interblade passage of each of the cascades. The calculation uses generalized space-time periodicity relations corresponding to the unsteady process of interest. The calculation is based on the direct numerical integration of the non-stationary gas dynamics equations with the use of the finite difference Godunov-Kolgan-Rodionov scheme of the second approximation order with respect to time and space. The calculation procedure includes the determination of the acoustic fields that are generated by the stage in the incident flow and in the flow behind it. The results of the calculations that illustrate the accuracy of the numerical solution and the capabilities of the method are presented.     

Variational approach to modeling and control of dynamic systems with distributed parameters

Controlled motions of a thin elastic rod are studied. A variational statement of the initial-boundary value problem is given based on the integral formulation of constitutive relations. An approach relying on the Ritz method and finite element technique is proposed to design the boundary control that moves the rod with certain accuracy to a given final state and minimizes the mean energy. The results of numerical simulation and the estimates of solution quality are presented and discussed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Harmonic polynomials and their application in solving problems of the mathematical theory of cracks

We consider new spectral relations for harmonic polynomials of two variables. The basic relations of this type are the integral relations in a disk, which make it possible to use these polynomials to construct the solution of certain singular equations of Newtonian potential type, of both first and second kind, given on the disk. By use of these results we determine the eigenvalues of certain two-dimensional integral equations of second kind.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 88–93.     

Approximate analytic solution of heat conduction problems with a mismatch between initial and boundary conditions

We consider a heat conduction problem for an infinite plate with a mismatch between initial and boundary conditions. Using the method of integral relations, we obtain an approximate analytic solution to this problem by determining the temperature perturbation front. The solution has a simple form of an algebraic polynomial without special functions. It allows us to determine the temperature state of the plate in the full range of the Fourier numbers (0≤F<∞) and is especially effective for very small time intervals.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

Singularly perturbed homotopy analysis method

In this paper we combined the homotopy analysis method (HAM) and the method of integral manifold (MIM) to investigate the problem of thermal explosion in two-phases polydisperse combustible mixtures of gas with fuel droplets. The size distribution of the fuel droplets is assumed to be continuous in the form of an exponential distribution and is found from the solution of the kinetic equation for the probability density function. The system of the polydisperse fuel spray takes into account the effects of the thermal radiation and convection. By applying the HAM and the MIM, we derived an analytical solution of the system and we compared our results with the numerical solutions.     

分数积分的一种数值计算方法及其应用

提出了一种只需要存储部分历史数据的分数积分的数值计算方法,并给出了误差估计。这种方法可对包含分数积分和分数导数的积分-微分方程进行较长时间的数值计算,克服了存储全部历史数据的困难,并能对计算误差进行控制。作为应用,给出了具有分数导数型本构关系的粘弹性Timoshenko梁的动力学行为研究的控制方程,利用分离变量法讨论梁在简谐激励作用下的动力响应,然后用新提出的数值方法对控制方程进行数值计算,数值计算结果和理论结果进行了比较,它们比较吻合。     

Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two-dimensional elastostatic problem under anti-plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.