Exact Analytic Solution of a Boundary Value Problem for the Falkner–Skan Equation

The Falkner–Skan equation, subject to appropriate physical boundary conditions arising from boundary layer theory, is exactly solved. The results obtained from this solution are compared with the numerical solution. The Blasius equation, subject to the same boundary conditions, is also solved exactly; the solution is compared with the earlier work on this equation. The analytic solution presented here agrees closely with the corresponding numerical results.     

螺旋槽球形轴承的承载能力

本文运用J·H·Vohr和C·H·T·Pan的理论和方法,建立了广义坐标系下的螺旋槽轴承的压力雷诺方程.然后在球轴承的边界条件下采用参数摄动法导出了动压螺旋槽球轴承润滑油膜的雷诺方程的近似解析解.由此,对各轴承槽型参数关于承载能力的影响作了计算和讨论,给出的最佳槽型参数与实验结果是一致的,与当前已发表的国内外资料相比较也是一致的.由于目前业已发表的文章均为计算机数值解,因此,本文对螺旋槽球轴承的特性研究提供了一个新的方法和途径.本文承中国科学院力学研究所林同骥,付仙罗同志及上海651研究所丁世德,蔡建中同志审阅,并提出了宝贵意见,作者谨在此表示衷心的感谢.     

Scattering of sound from point sources by multiple circular cylinders using addition theorem and superposition technique

In this study, we use the addition theorem and superposition technique to solve the scattering problem with multiple circular cylinders arising from point sound sources. Using the superposition technique, the problem can be decomposed into two individual parts. One is the free‐space fundamental solution. The other is a typical boundary value problem (BVP) with specified boundary conditions derived from the addition theorem by translating the fundamental solution. Following the success of null‐field boundary integral formulation to solve the typical BVP of the Helmholtz equation with Fourier densities, the second‐part solution is easily obtained after collocating the observation point exactly on the real boundary and matching the boundary condition. The total solution is obtained by superimposing the two parts which are the fundamental solution and the semianalytical solution of the Helmholtz problem. An example was demonstrated to validate the present approach. The parameter study of size and spacing between cylinders are addressed. The results are well compared with the available theoretical solutions and experimental data. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Saturated-unsaturated groundwater modeling using 3D Richards equation with a coordinate transform of nonorthogonal grids

Saturated-unsaturated flow under a complex terrain is usually solved using the Richards equation. Finite difference or finite volume methods are commonly employed for discretization of Richards equation because of simplicity of coding. Complex subsurface boundary geometries lead to nonorthogonal grids in curvilinear grid systems, which leads to difficulty in discretization and mesh generation. This paper develops a vertical coordinate transform, enabling a computational domain regular in the vertical direction. As a result, the grid of curvilinear surfaces can be successfully transformed to a computational grid that allows solution of the Richards equation with efficient computation and simpler coding. The anisotropic Richards equation in the Cartesian coordinate system is transformed to the equation in the arbitrary coordinate system and further expressed as a form appropriate to the orthogonal coordinate system. The generalized third boundary condition is transformed to a form suited to the orthogonal coordinate system. The finite volume method is used to solve the Richards equation in the orthogonal coordinate system. Four cases are used to validate the present orthogonal coordinate system. The computational results from the orthogonal coordinate system are in good agreement with the results from Ansys Fluent solved in a Cartesian coordinate system for the subsurface flow case. For the coupled case of hill slopes, a good agreement between the computational results and the experimental data is obtained. The present results for V-titled catchment and slab case accord well with the results obtained from HydroGeoSphere and PAWS. The present algorithm can improve grid generation for solution of Richards equation in a hydrological model for a complex domain.     

On some solutions of Chapman-Kolmogorov equation for discrete-state Markov processes with continuous time

The nonlinear equation mentioned in the title is the basic one in the theory of Markov processes. In the case of a discrete-state process, its solution is given by the transition probability function. Usually, solving this equation amounts to solving a linear equation. In 1932, S. N. Bernstein posed the problem of direct determination of the solution. In 1961, such solutions were given in terms of bilinear series by O. V. Sarmanov for stationary continuous-state Markov processes. In 2007, several solutions were obtained by the author in terms of generalized bilinear series without placing Sarmanov’s restrictions. In this paper, our results are extended to discrete-state processes. Two solutions of the Chapman-Kolmogorov equation are derived by means of reducing it to some functional equation. The solutions are represented in the form of a bilinear sum and its generalizations, each term of the sum being proportional to the product of two orthogonal functions. The results obtained are illustrated by two-state processes, which exemplify the assertions derived in this paper. Another example is used to show that the Chapman-Kolmogorov equation has a solution which is devoid of probabilistic sense.     

An improved boundary element method for the charge density of a thin electrified plate in ℝ3

A weakly singular integral equation of the first kind on a plane surface piece Γ is solved approximately via the Galerkin method. The determination of the solution of this integral equation (with the single-layer potential) is a classical problem in physics, since its solution represents the charge density of a thin, electrified plate Γ loaded with some given potential. Using piecewise constant or piecewise bilinear boundary elements we derive asymptotic estimates for the Galerkin error in the energy norm and analyse the effect of graded meshes. Estimates in lower order Sobolev norms are obtained via the Aubin–Nitsche trick. We describe in detail the numerical implementation of the Galerkin method with both piecewise-constant and piecewise-linear boundary elements. Numerical experiments show experimental rates of convergence that confirm our theoretical, asymptotic results.     

Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field

In this paper the influence of the gravity field, relaxation times and initial stress on propagation of Rayleigh waves in an orthotropic magneto-thermoelastic solid medium has been investigated. The solution of the more general equations are obtained for thermoelastic coupling by Helmoltz’s theorem. The frequency equation which determines Rayleigh wave velocity have been obtained. Many special cases are investigated from the present problem. Numerical results analyzing the frequency equation are obtained and presented graphically. Relevant results of previous investigations are deduced as special cases from these results. The results indicate that the effect of initial stress, magnetic field and gravity field are very pronounced.     

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.     

A Smoothing Method of Global Optimization that Preserves Global Minima

A new smoothing method of global optimization is proposed in the present paper, which prevents shifting of global minima. In this method, smoothed functions are solutions of a heat diffusion equation with external heat source. The source helps to control the diffusion such that a global minimum of the smoothed function is again a global minimum of the cost function. This property, and the existence and uniqueness of the solution are proved using results in theory of viscosity solutions. Moreover, we devise an iterative equation by which smoothed functions can be obtained analytically for a class of cost functions. The effectiveness and potential of our method are then demonstrated with some experimental results.     

A computational meshless method for the generalized Burger’s–Huxley equation

A numerical solution of the generalized Burger’s–Huxley equation, based on collocation method using Radial basis functions (RBFs), called Kansa’s approach is presented. The numerical results are compared with the exact solution, Adomian decomposition method (ADM) and Variational iteration method (VIM). Highly accurate and efficient results are obtained by RBFs method. Excellent agreement with the exact solution is observed while better (or same) accuracy is obtained than other numerical schemes cited in this work.     

Nonlinear Schrödinger Equations and Their Spectral Semi-Discretizations Over Long Times

Cubic Schrödinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both the solution of the nonlinear Schrödinger equation as well as the solution of the semi-discretized equation the actions of the linear Schrödinger equation are approximately conserved over long times. This also allows us to show approximate conservation of energy and momentum along the solution of the semi-discretized equation over long times. These results are obtained by analyzing a modulated Fourier expansion in time. They are valid in arbitrary spatial dimension.     

Boundary function method to find the asymptotic solution of the plasma—sheath equation

We consider the asymptotic solution of the Tonks—Langmuir integro-different equation with an Emmert kernel, which describes the behavior of the potential both inside the main plasma volume and in a thin boundary layer. Equations of this type are singularly perturbed due to the small coefficient at the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the regular expansion series and in the boundary function expansion. The equation for the first coefficient of the regular series has only a trivial solution. Second-order differential equations are obtained for the first two boundary functions. The equation for the first boundary function is solved numerically on a discrete grid with locally uniform spacing. An approximate analytical expression for the first boundary function is obtained from the linearized equation. This solution adequately describes the behavior of the potential on small distances only. __________ Translated from Prikladnaya Matematika i Informatika, No. 19, pp. 21–40, 2004.     

EXISTENCE OF POSITIVE PERIODIC SOLUTION TO A CLASS OF NONAUTONOMOUS DIFFERENTIAL EQUATION WITH IMPULSES AND DELAY

In this paper, by fixed point theorem of Krasnoselskii, we study the positive pe- riodic solution to a class of nonautonomous differential equation with impulses and delay. Firstly, definition of periodic solution and some lemmas are stated. Then some results on the existence of positive periodic solution to the equation are obtained.     

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green’s function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.     

On the Solution of a System of Hamilton–Jacobi Equations of Special Form

The paper is concerned with the investigation of a system of first-order Hamilton–Jacobi equations. We consider a strongly coupled hierarchical system: the first equation is independent of the second, and the Hamiltonian of the second equation depends on the gradient of the solution of the first equation. The system can be solved sequentially. The solution of the first equation is understood in the sense of the theory of minimax (viscosity) solutions and can be obtained with the help of the Lax–Hopf formula. The substitution of the solution of the first equation in the second Hamilton–Jacobi equation results in a Hamilton–Jacobi equation with discontinuous Hamiltonian. This equation is solved with the use of the idea of M-solutions proposed by A. I. Subbotin, and the solution is chosen from the class of multivalued mappings. Thus, the solution of the original system of Hamilton–Jacobi equations is the direct product of a single-valued and multivalued mappings, which satisfy the first and second equations in the minimax and M-solution sense, respectively. In the case when the solution of the first equation is nondifferentiable only along one Rankine–Hugoniot line, existence and uniqueness theorems are proved. A representative formula for the solution of the system is obtained in terms of Cauchy characteristics. The properties of the solution and their dependence on the parameters of the problem are investigated.     

State of stress and strain of brake-class parachutes

The article describes the special features of the operation of brake-class parachutes. It substantiates the possibility of replacing the solution of the dynamic problem of opening of a brake parachute by the static calculation of its state of stress and strain (SSS). The derivation of the equation of motion of a soft carcassed shell is based on the finite element method. The steady-state solution is obtained by the method of adjustment. As an example the results of calculation of the characteristics of the SSS of a cross-shaped brake parachute are presented. It is shown that in the zone of its lower edge considerable concentrations of tension arise in the tissue, the tissue gradually joins the operation of the carcass; the transverse carcass is unsubstantially loaded. The results of the calculations agree with the experimental data.Translated from Dinamicheskie Sistemy, No. 5, pp. 32–36, 1986.     

On the solution of Dirichlet problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the second type

Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

The role of the Fox–Wright functions in fractional sub-diffusion of distributed order

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin–Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox–Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.     

Modeling and analysis of the forced Fisher equation

The Fisher equation with inhomogeneous forcing is considered in this paper. First, a forced Fisher equation and boundary conditions are derived. Then, the existence of a local solution for the forced equation with a homegeneous Dirichlet condition is proved by Galerkin's method. Next, a maximum principle is established and the existence of a global solution is obtained as a consequence of the maximum principle. Finally, generalizations of the results to cases of less regular forces are discussed.