An exact methodology for solving nonlinear diffusion equations based on integral transforms

This paper develops a new methodology for the solution of nonlinear diffusion equations. The solution technique is based on integral transforms and leads to exact numerical results. We apply the formal methodology to the problem of one-dimensional transient heat conduction. A new form of the heat equation is developed using a generalized expression for temperature-dependent thermal conductivity, based on a power-series expansion, for the three standard orthogonal coordinate systems. The resulting form of the heat equation suggests that the finite integral transform technique may reduce the dimensionality of the heat equation prior to the initiation of any numerical procedure. An example in a slab with linearly varying thermal conductivity is shown to produce exact results for the temperature distribution.     

基于摄动法的压敏储层非线性渗流问题研究

研究地层压力变化对渗流特征的影响,对低渗透储层、碳酸盐岩储层,或其他致密性储层的油气开采和储层改造都具有重要意义.在考虑渗流压力梯度平方项存在的前提下,运用摄动法求解的相关理论,将渗透率随压力的变化融入到渗流问题的求解过程中,有效地求解了该类压敏型储层的非线性渗流问题.结果表明,在实际应用中,尤其是在储层压敏性较弱的情况下,可考虑直接用0阶摄动解即可满足较好的计算精度;储层的压敏性越强,越适宜通过用摄动解的修正,来达到精确求解的目的.     

On generalized Stieltjes-Wigert and related orthogonal polynomials

The Charlier, Wall, and generalized Stieltjes-Wigert polynomials are characterized by a property involving the concept of kernel polynomials. This characterization leads to consideration of a certain functional equation satisfied by solutions of the associated Stieltjes moment problem. All distribution functions which satisfy this functional equation are found up to singular functions. This yields new distribution functions, both discrete and absolutely continuous, with respect to which generalized Stieltjes-Wigert polynomials are orthogonal.     

Particular solutions of the linearized Boltzmann equation for a binary mixture of rigid spheres

Particular solutions that correspond to inhomogeneous driving terms in the linearized Boltzmann equation for the case of a binary mixture of rigid spheres are reported. For flow problems (in a plane channel) driven by pressure, temperature, and density gradients, inhomogeneous terms appear in the Boltzmann equation, and it is for these inhomogeneous terms that the particular solutions are developed. The required solutions for temperature and density driven problems are expressed in terms of previously reported generalized (vector-valued) Chapman–Enskog functions. However, for the pressure-driven problem (Poiseuille flow) the required particular solution is expressed in terms of two generalized Burnett functions defined by linear integral equations in which the driving terms are given in terms of the Chapman–Enskog functions. To complete this work, expansions in terms of Hermite cubic splines and a collocation scheme are used to establish numerical solutions for the generalized (vector-valued) Burnett functions.     

Particular solutions of the linearized Boltzmann equation for a binary mixture of rigid spheres

Particular solutions that correspond to inhomogeneous driving terms in the linearized Boltzmann equation for the case of a binary mixture of rigid spheres are reported. For flow problems (in a plane channel) driven by pressure, temperature, and density gradients, inhomogeneous terms appear in the Boltzmann equation, and it is for these inhomogeneous terms that the particular solutions are developed. The required solutions for temperature and density driven problems are expressed in terms of previously reported generalized (vector-valued) Chapman–Enskog functions. However, for the pressure-driven problem (Poiseuille flow) the required particular solution is expressed in terms of two generalized Burnett functions defined by linear integral equations in which the driving terms are given in terms of the Chapman–Enskog functions. To complete this work, expansions in terms of Hermite cubic splines and a collocation scheme are used to establish numerical solutions for the generalized (vector-valued) Burnett functions.     

考虑二次梯度项影响的非线性不稳定渗流问题的精确解

考虑了二次梯度项影响的非线性径向流动问题的无限大地层和有界地层渗流模型．在井底定流量和定压生产时,对无限大地层及有界地层(包括封闭和定压地层)六种情况,利用广义Weber变换和广义Hankel变换求得了实空间的解析解,分析了非线性压力解与线性压力解的差异,发现在晚时段其差异可达8%以上．因此在试井长时要考虑二次梯度项的影响．     

Integrability and Self-Similarity in Transient Stimulated Raman Scattering

Summary. The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying self-similar solution which satisfies a particular case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields. Received December 10, 1996; second revision received October 10, 1997; final revision received January 20, 1998     

The closure of the sheaf of trajectories of a linear control system with integral constraints

We consider a linear system with discontinuous coefficients controlled by a parameter under an integral constraint imposed on the control resource. It is well known that in such problems the closure of the sheaf of trajectories that correspond to ordinary controls (piecewise constant or measurable functions) coincides with the sheaf of trajectories in a generalized problem, where for generalized controls one uses finite additive measures of bounded variation. Therewith the closure is defined in the topology of pointwise convergence, because the limit elements (the generalized trajectories) may be discontinuous functions. In this paper we prove that any generalized trajectory can be approximated by a sequence of ordinary solutions to the initial system. We propose a concrete technique for constructing such sequences.     

Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.     

Monotonicity and finite element methods for hyperbolic initial value problems

A well known theórem about super- and subfunctions for the solution of hyperbolic initial value problems constructs differentiable functions as upper and lower bounds (see Walter [1], 21 XIII). The proof can be done by transforming the differential equation problem into a set of integral equations, using the monotonicity-properties of the arising integral operators. This proof needs an integral representation for twice differentiable functions. It is shown that this proceeding can be generalized to get upper and lower bounds in terms of finite element functions. To do this, we give an integral representation for continuous, piecewise differentiable functions, including the discontinuities of their derivatives. Then the generalization of the classical proof yields interface conditions for the finite element functions. Finally, it is demonstrated how to realize numerically these conditions.     

Analog of the cauchy function for a generalized equation with several deviations of the argument

We define a special multiplication of function series (skew multiplication) and a generalized Riemann-Stieltjes integral with function series as integration arguments. The generalized integrals and the skew multiplication are related by an integration by parts formula. The generalized integrals generate a family of linear generalized integral equations, which includes a family (represented in integral form via the Riemann-Stieltjes integral) of linear differential equations with several deviating arguments. A specific feature of these equations is that all deviating functions are defined on the same closed interval and map it into itself. This permits one to avoid specifying the initial functions and imposing any additional constraints on the deviating functions. We present a procedure for constructing the fundamental solution of a generalized integral equation. With respect to the skew multiplication, it is invertible and generates the product of the fundamental solution (a function of one variable) by its inverse function (a function of the second variable). Under certain conditions on the parameters of the equation, the product has all specific properties of the Cauchy function. We introduce the notion of adjoint generalized integral equation, obtain a representation of solutions of the original equation and the adjoint equation in generalized integral Cauchy form, and derive sufficient conditions for the convergence of solutions of a pair of adjoint equations.     

A posteriori error estimates for the generalized Stokes problem

The paper concerns a posteriori estimates of functional type for the difference between exact and approximate solutions to a generalized Stokes problem. The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem using the method suggested by the first author. The estimates obtained can be classified into two types. Estimates of the first type are valid only for solenoidal functions, while estimates of the second type are applicable for any functions that belong to the energy space of the respective problem and satisfy the boundary conditions. In the second case, the estimates include an additional penalty term with a multiplier defined by the constant in the Ladyzhenskaya-Babuška-Brezzi condition. It is proved that a posteriori estimates for the velocity field yield computable estimates of the difference between exact and approximate pressure functions in the L₂-norm. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires to solve only finite-dimensional problems. Bibliography: 34 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 89–101.     

缝洞型碳酸盐岩油藏注水驱油提高采收率研究

Y油田为缝洞型碳酸盐岩油藏,油藏复杂的地质特征决定了其开发模式有别于常规砂岩油藏.为了有效提高Y油田采收率,利用Eclipse数值模拟软件建立了地质模型,对水平井长度、射孔间距及酸压缝长等油藏产能影响因素与转注时机、注采比以及关井时间等注采参数进行了优化.结果认为:对于Y缝洞型碳酸盐岩油藏,水平井水平段长度为500m、射孔间距为20m或酸压规模为100m时,累计产油量可达到最大值;压力降至废弃压力时进行转注,可获得最大采收率;周期注采比等于1时,周期时效与累计增油量达到最大值;关井时间为8天时,采收率提高幅度最大.     

计及液面波动和液体可压缩性时部分潜入水中椭圆柱体的扭转振动

本文研究部分潜入水中椭圆柱体的扭转振动问题,同时计及了液面波动和液体可压缩性对椭圆柱体扭转振动的影响。利用马休(Mathieu)级数以及一组广义三角级数的正交完备性,导出了柱水耦联扭振振型函数和频率方程的精确解析解,可利用非线性代数方程的求根方法,数值解出各阶频率参数。     

On ∑-lossless transfer functions and related questions

This paper is concerned with a systematic approach to the properties of ∑-lossless rational transfer functions in the discrete as well as in the continuous time case. As a result, a unifying framework is revealed where several known results fit naturally. Special attention is given to the embedding problem of the Lyapunov equation in view of its direct application to generalized Levinson algorithms.     

Solutions of Riccati-Abel equation in terms of third order trigonometric functions

Solutions of the generalized Riccati equations with third order nonlinearity, named as Riccati-Abel equation, are expressed via third order trigonometric functions. It is shown, as the ordinary Riccati equation, also the Riccati-Abel equation has a relationship with a linear differential equations. A summation formula for solutions of Riccati-Abel equation is established. Possible applications of this formula in the generalized dynamics is outlined. The method admits an extension to the case of generalized Riccati equations with any order of nonlinearity     

Distributional solutions to a generalized wave equation for gravity waves on deep water

The equations governing the linearized small amplitude approximation for gravity waves on deep water can be reformulated by the introduction of a cross-surface differential operator, H, which acts like a square-root of the two-dimensional Laplacian. This yields a single scalar equation for the amplitude of the wave-like motion off a horizontal static surface resulting in a mixed initial and boundary value problem for the wave operator, ∂_tt + c²H. The pressure impulse response for an unperturbed static fluid will be calculated via a formal eigenfunction expansion and it will be shown that this yields a distributional solution. Then, the mixed problem will be generalized to allow for distributional data where the initial data is injected into the non-homogeneous term. By employing eigenfunction representations for distributions with compact support it will be shown that a formal eigenfunction expansion also yields a distributional solution to this generalized mixed problem.     

The construction of Poincaré-Chetayevand Smale bifurcation diagrams for conservative non-holonomic systems with symmetry

The problem of the existence of first integrals which are linear functions of the generalized velocities (momenta and quasi-velocities) is discussed for conservative non-holonomic Chaplygin systems with symmetry, as well as methods for investigating the existence, stability, and bifurcation of the steady motions of such systems. These methods are based on the classical methods of Routh-Salvadori, Poincaré-Chetayev, and Smale, but unlike the latter they do not require a knowledge of the explicit form of the linear integrals. The general conclusions are illustrated by the example of the problem of an ellipsoid of revolution moving on an absolutely rough horizontal surface. It is shown how in this case numerical techniques can be used to construct the Poincaré-Chetayev diagram — a surface in the space of generalized coordinates and constants of linear first integrals corresponding to motions in which the velocities of the non-cyclic coordinates vanish, while those of the cyclic coordinates are constant, and the Smale diagram — a surface in the space of constants of linear first integrals and the energy integral corresponding to these motions.     

Predicting stochastic characteristics of generalized eigenvalues via a novel sensitivity-based probability density evolution method

This paper proposes a novel numerical method for predicting the probability density function of generalized eigenvalues in the mechanical vibration system with consideration of uncertainties in structural parameters. The eigenproblem of structural vibration is presented by first and the sensitivity of generalized eigenvalues with respect to structural parameters can be derived. The probability density evolution method is then developed to capture the probability density function of generalized eigenvalues considering uncertain material properties. Within the proposed method, the probability density evolution equation for the generalized eigenvalue problem is established accounting for the sensitivity of generalized eigenvalues with respect to structural parameters. A new variable which connects generalized eigenvalues to structural parameters is then introduced to simplify the original probability density evolution equation. Next, the simplified probability density evolution equation is solved by using the finite difference method with total variation diminishing schemes. Finally, the probability density function as well as the second-order statistical quantities of generalized eigenvalues can be predicted. Numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method within significantly less computation time and the coefficients of variation of uncertain parameters as well as the total number of them have remarkable effects on stochastic characteristics of generalized eigenvalues.