一种带有奇异核的Volterra第二种积分方程的解

本文提供了一种解带有特定奇异核为1/(s-t)~a(0<a<1) 的 Volterra 第2种积分方程的方法。主要是通过对方程进行拉氏变换、函数的幂级数展开等过程,求得未知函数的拉氏变换表达式,然后对其进行反变换,即得所求函数。最后给出了此解的证明及 a=1/2的特例。

Cracked elastic substrate strip with functionally graded coating under thermal-mechanical loading

This paper investigates the functionally graded coating bonded to an elastic strip with a crack under thermal- mechanical loading. Considering some new boundary conditions, it is assumed that the temperature drop across the crack surface is the result of the thermal conductivity index which controls heat conduction through the crack region. By the Fourier transforms, the thermal-elastic mixed boundary value problems are reduced to a system of singular integral equations which can be approximately solved by applying the Chebyshev polynomials. The numerical computation methods for the temperature, the displacement field and the thermal stress intensity factors （TSIFs） are presented. The normal temperature distributions （NTD） with different parameters along the crack surface are analyzed by numerical examples. The influence of the crack position and the thermal-elastic non- homogeneous parameters on the TSIFs of modes I and 11 at the crack tip is presented. Results show that the variation of the thickness of the graded coating has a significant effect on the temperature jump across the crack surfaces when keeping the thickness of the substrate constant, and the thickness of functionally graded material （FGM） coating has a significant effect on the crack in the substrate. The results can be expected to be used for the purpose of gaining better understanding of the thermal-mechanical behavior of graded coatings.

Wave Interaction with Dual Circular Porous Plates

In this paper we have investigated the reflection and the transmission of a system of two symmetric circular-arc-shaped thin porous plates submerged in deep water within the context of linear theory. The hypersingular integral equation technique has been used to analyze the problem mathematically. The integral equations are formulated by applying Green＇s integral theorem to the fundamental potential function and the scattered potential function into a suitable fluid region, and then using the boundary condition on the porous plate surface. These are solved approximately using an expansion-cure-collocation method where the behaviour of the potential functions at the tips of the plates have been used. This method ultimately produces a very good numerical approximation for the reflection and the transmission coefficients and hydrodynamic force components. The numerical results are depicted graphically against the wave number for a variety of layouts of the arc. Some results are compared with known results for similar configurations of dual rigid plate systems available in the literature with good agreement.

New Explicit and Exact Travelling Wave Solutions for Burgers-Kolmogorov-Petrovskii-Piscounov Equations

In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.

Analysis of the One Dimensional Steady State Semiconductor Multijunction Device Model with the Avalanche Effect *

In this paper we consider the stationary multijunction device model with the avalanche effect. Using the singular perturbation method, an approximation to the current voltage curve is obtained. The cause and the condition for the occurrence of saturation current is analyzed. Especially, it is pointed that the avalanche effect is responsible for the blowing up of the saturation current. We prove the existence of multiple steady state solution when the ionization rate is relatively small. Finally, some numerical examples are presented to show the reliability of the theoretical results.

Analysis of unsteady supercavitating flow around a wedge

Supercavitating flow around a slender symmetric wedge moving at variable velocity in static fluid has been studied. Singular integral equation for the flow has been founded through distributing the sources and sinks on the symmetrical axis. The supereavity length at each moment is determined by solving the singular integral equation with finite difference method. The supercavity shape at each moment is obtained by solving the partial differential equation with variable coefficient. For the case that the wedge takes the impulse and uniformly variable motion, numerical results of time history of the supercavity length and shape are presented. The calculated results indicate that the shape and the length of the supercavity vary in a similar way to the case that the wedge takes variable motion, and there is a time lag in unsteady supercavitating flow induced by the variation of wedge velocity.

The Nonlinear Predator-Prey Singularly Perturbed Robin Initial Boundary Value Problems for Reaction Diffusion System

The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.

A Quadrature Approach for N-Collinear Crack Problem in an Orthotropic Strip

This study presents the determination of the stress intensity factors （SIFs） at the edges of the cracks in an elastic strip weakened by N-collinear cracks. The problem of an orthotropic elastic strip is reduced to a system of Cauchy type singular integral equations. The system of singular integral equations is approached by a Quadrature technique. Under two different loading conditions, the results are obtained for the different cases of crack numbers. The resistance of the strip is examined by considering the orthotropic properties of the strip material. Finally, the crack interactions are clarified during the analysis.

CAUCHY PROBLEM FOR A NONLINEAR SINGULAR INTEGRAL－DIFFERENTIAL EVOLUTION SYSTEM

We study the Cauchy problem for a nonlinear evolution system with singularintegral differential terms.By means of some a priori estimates of the solution and theLeray-Schauder's fixed point theorem, we prove the existence and the uniqueness theoremsof the generalized global solution of the mentioned problem.

Analysis of Singularity for Reducible Quasi-linear Hyperbolic Systems

In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.

On a Singular Solution in Higgs Field （Ⅲ）： Condensates and Representation of Certain f0 Mesons＇ Masses

Recently the mass and the basic structure of SM Higgs boson （H^0） have been discussed by the author with obtaining asymptotic solution for the equation of motion of nonlinear Klein-Gordon type partial differential equation. In this paper, we will treat with above in mind： masses of glueball （GB） of ground state and of certain fo mesons, ur, Higgs boson （ur-H^0） which will consist of a number of GBs and/or fo above for respective fullerene structure, a representation of these fo mesons＇ masses by masses of n octet and GB, and transformation of ur-H^0 into H^0.

Integral Representation of the Manifold Solution for New Class of the Volterra Type Integral Equation with a Boundary Singularity in the Case, When Kernel Contain Logarithmic Singularity and its Power

In this work we suggestion new methods investigation the model Volterra type integral equation with logarithmic singularity, kernel which consisting from composition polynomial function with logarithmic singularity and function with singular point. The problem investigation this type integral equation at n = 2m reduce to problem investigate the Volterra type integral equation （1） for n = 2 the theory for which was constructed in [2]. In this work, we investigation integral equation （1） at = 2m ＋ 1 In this case, we investigate integral equation （1） reduction it＇s to m integral equation type [2] φ（x）＋∫xa[p1＋p2 ln（x-a/t-a）]φ（t）/t-a dt=f（x）and one the following integral equation [1] ω（x）＋p3∫xω（t）/ a t-adt=g（x）.

Singular Perturbation of a Class of Dirichlet Exterior Problems for Elliptic Equations

In this paper, we consider a class of singularly perturbed Dirichlet exterior problems for elliptic equations. Under the appropriate conditions we construct the formally asymptotic solution of the problem described. Using differential inequaltiy theory we prove the existence of the solution of original problem and the uniforly validity of the formal solution.

The Application of the Fixed-point Theory in Jacketed Solution of Singularly Perturbed Nonlinear Problem

In this paper,the fixed_point theorem is used to estimated an asymptotic solution of initial value problems for a class of third nonlinear differential equations which has double initial_layer properties. We obtain the uniformly valid asymptotic expansion of any orders including boundary layers.

Existence of Solutions to a Class of Higher-Order Singular Boundary Value Problem for One-Dimensional p-Laplacian

This paper deals with the existence of positive solutions for the problem {（Фp（x^（n-1）（t）））′＋f（t,x,…,x^（n-1）=0,0〈t〈1, x^（i）（0）=0,0≤i≤n-3, x^（n-2）（0）-B0（x^（n-1）（0））=0,x^（n-2）（1）＋B1（x^（x-1）（1））=0, where Фp（s） = ｜s｜^p-2s, p 〉 1. f may be singular at x^（i） = 0, i = 0,...,n- 2. The proof is based on the Leray-Schauder degree and Vitali＇s convergence theorem.

Some progress in spectral methods Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.

PROPERTIES OF THE BOUNDARY FLUX OF A SINGULAR DIFFUSION PROCESS

The authors study the singular diffusion equationwhere Ω(?)Rn is a bounded domain with appropriately smooth boundary δΩ, ρ(x) = dist(x,δΩ), and prove that if α≥p-1, the equation admits a unique solution subject only to a given initial datum without any boundary value condition, while if 0 <α< p - 1, for a given initial datum, the equation admits different solutions for different boundary value conditions.

Positive Solutions of Singular Boundary Value Problems of Fourth-Order Differential Equations

The singular boundary value problem{φ^（4）（x）-h（x）f（φ（x））=0,0〈x〈1, φ（0）=φ（1）=φ′（0）=φ′（1）=0.is considered under some conditions concerning the first eigenvaiues corresponding to the relevant linear operators, where h（x） is allowed to be singular at both x = 0 and x = 1. The existence results of positive solutions are obtained by means of the cone theory and the fixed point index.

ON SUFFICIENT AND NECESSARY OF EXISTENCE FOR A CLASS OF SINGULAR OPTIMAL STOCHASTIC CONTROL

We study a class of discounted models of singular stochastic control. In thiskind of models, not only the structure of cost function has been extended to some general type, butalso the state can be represented as the solution of a class of stochastic differential equationswith nonlinear drift and diffusion term. By the various methods of stochastic analysis, we derivethe sufficient and necessary conditions of the existence of optimal control.

Stress fields caused by a dislocation in an anisotropic 3-layer system

We investigated the stress fields caused by a dislocation in an anisotropic 3-layer system. Based on the image method, the original 3-layer system is firstly decomposed into three infinite homogenous systems. The image dislocation densities used as unknowns are then strategically distributed in order to satisfy the boundary conditions. The resulting governing equations are singular Cauchy integral ones. Removing the singular terms yields non-linear Fredhom integral equations of the second kind. The obtained stress fields satisfy the boundary conditions, i.e., the traction free condition on the free surface and continuous conditions across the interfaces. Also, a comparison with previous results is made and good agreement is achieved. Numerical investigations show that under the plain strain condition, layer thickness and dislocation position play stronger roles in the stress fields than crystallographic orientation, and these effects more significantly affect the stress fields caused by an edge dislocation than by a screw dislocation.