Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape

A general method is presented for the rigorous solution of Eshelby’s problem concerned with an arbitrary shaped inclusion embedded within one of two dissimilar elastic half-planes in plane elasticity. The bonding between the half-planes is considered to be imperfect with the assumption that the interface imperfections are uniform. Using analytic continuation, the basic boundary value problem is reduced to a set of two coupled nonhomogeneous first-order differential equations for two analytic functions defined in the lower half-plane which is free of the thermal inclusion. Using diagonalization, the two coupled differential equations are decoupled into two independent nonhomogeneous first-order differential equations for two newly defined analytic functions. The resulting closed-form solutions are given in terms of the constant imperfect interface parameters and the auxiliary function constructed from the conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. The method is illustrated using several examples of an imperfect interface. In particular, when the same degree of imperfection is realized in both the normal and tangential directions between the two half-planes, a thermal inclusion of arbitrary shape in the upper half-plane does not cause any mean stress to develop in the lower half-plane. Alternatively, when the imperfect interface parameters are not equal, then a nonzero mean stress will be induced in the lower half-plane by the thermal inclusion of arbitrary shape in the upper half-plane. Detailed results are presented for the mean stress and the interfacial normal and shear stresses caused by a circular and elliptical thermal inclusion, respectively. Results from these calculations reveal that the imperfect bonding condition has a significant effect on the internal stress field induced within the inclusion as well as on the interfacial normal and shear stresses existing between the two half-planes especially when the inclusion is near the imperfect interface.     

The Cauchy Problem for the Kadomtsev–Petviashili II Equation with Nondecaying Data along a Line

The initial value problem for the Kadomstev–Petviashvili II (KPII) equation is considered with given data that are nondecaying along a line. The associated direct and inverse scattering of the two-dimensional heat equation is constructed. The direct problem is formulated in terms of a bounded Green's function. The inverse data are decomposed into scattering data along the line and     data from the decaying portion of the potential. The solution of the KPII equation is then obtained via coupled linear integral equations.     

Rational inversion of the Laplace transform

This paper studies new inversion methods for the Laplace transform of vector-valued functions arising from a combination of A-stable rational approximation schemes to the exponential and the shift operator semigroup. Each inversion method is provided in the form of a (finite) linear combination of the Laplace transform of the function and a finite amount of its derivatives. Seven explicit methods arising from A-stable schemes are provided, such as the Backward Euler, RadauIIA, Crank-Nicolson, and Calahan scheme. The main result shows that, if a function has an analytic extension to a sector containing the nonnegative real line, then the error estimate for each method is uniform in time.     

The inverse Penrose transform of a solution to the Maxwell-Dirac-Weyl field equations

An explicit family of solutions to the nonlinear coupled Maxwell-Dirac-Weyl equations in Minkowski space is presented. The abstract results of Henkin and Manin (Phys. Lett. B, 95 (1980), 405–408) show that these solutions are equivalent by the Penrose transform to a coupled system of cohomology classes and a complex line bundle on ambitwistor space, the space of null lines in Minkowski space. The explicit inverse Penrose transform of this family of solutions is computed giving explicit expressions for the line bundle (transform of the vector potential), the obstruction to extension (transform of the charge), and the two cohomology classes (transform of the Dirac-Weyl coupled spinor fields).     

On a special function used in the description of electromagnetic surface waves

The tangential component of the electric field of a surface wave at any distance from the transmitting antenna lying in the interface plane of two homogeneous media can be represented in terms of a function of two complex variables Ŵ(q,ξ) for arbitrary parameters of the interface. In this paper, representations of the function Ŵ(q,ξ) in the form of series are given that allow one to quickly calculate the values of Ŵ(q,ξ) and to investigate the analytic properties of this function. The dependence of the field of the surface wave on time is determined using the inverse Laplace transform, where the path of integration is chosen in such a way that the integrand rapidly decreases at infinity, which drastically improves the computation speed compared with the method based on the Fourier transform.     

Three-dimensional Green's functions in anisotropic magneto-electro-elastic bimaterials

In this paper, we derive three-dimensional Green's functions in anisotropic magneto-electro-elastic full space, half space, and bimaterials based on the extended Stroh formalism. While in the full space, the Green's functions are obtained in an explicit form, those in the half space and bimaterials are expressed as a sum of the full-space Green's function and a Mindlin- type complementary part, with the latter being evaluated in terms of a regular line integral over [0, p][0, \pi]. Despite the complexity involved, the current Green's function expressions are surprisingly simple. Furthermore, the piezoelectric, piezomagnetic, and purely elastic Green's functions can all be obtained from the current Green's functions by setting simply the appropriate material coefficients to zero. A special material case, to which the extended Stroh formalism cannot be applied directly, has also been identified.¶Simple numerical examples are presented for Green's functions in full space, half space, and bimaterials with fully coupled and uncoupled anisotropic magneto-electro-elastic material properties.For given material properties and fixed source and field points, the effect of magneto-electro-elastic coupling on the Green's function is discussed. In particular, we observed that magneto-electro-elastic coupling could significantly alter the magnitude of certain Green's displacement and stress components, with difference as high as 45% being noticed. This result is remarkable and should be of great interest in the material analysis and design.     

旋波媒质中谱域电并矢Green函数的分解

介绍了一种推导无耗、互易和无界旋波媒质中谱域并矢Green函数表达式的新方法·　这种方法以Hemholtz定理以及并矢Diracδ函数的无散和无旋分解为基础,首先将电矢量的并矢Green函数方程分解成无散电矢量的并矢Green函数方程和无旋电矢量的并矢Green函数方程,然后经Fourier变换导出了旋波媒质中谱域电并矢Green函数的无散分解表达式和无旋分解表达式·　用这种方法推导旋波媒质中并矢Green函数就可以避免必须用波场的分解法和并矢Green函数的本征函数展开法·     

Preconditioning the Poincaré-Steklov operator by using Green's function

This paper is concerned with the Poincaré-Steklov operator that is widely used in domain decomposition methods. It is proved that the inverse of the Poincaré-Steklov operator can be expressed explicitly by an integral operator with a kernel being the Green's function restricted to the interface. As an application, for the discrete Poincaré-Steklov operator with respect to either a line (edge) or a star-shaped web associated with a single vertex point, a preconditioner can be constructed by first imbedding the line as the diameter of a disk, or the web as a union of radii of a disk, and then using the Green's function on the disk. The proposed technique can be effectively used in conjunction with various existing domain decomposition techniques, especially with the methods based on vertex spaces (from multi-subdomain decomposition). Some numerical results are reported.

     

Solution to a Class of Coupled Linear Partial Differential Equations

The solution to a coupled system of partial differential equationsinvolving a general linear time-independent operator L is presented.Examples of these equations include coupled diffusion equationsor coupled convection–dispersion equations. The solutionconsists of a convolution of the Green's function appropriatefor the operator L and a function independent of the operatorL. The method enables one to write software to calculate thesolution to a wide range of problems. The change of solutionupon changing the problem often only involves a substitutionof the Green's function. A specific example of physical significanceis given.     

Electrical coupling in proton exchange membrane fuel cell stacks: mathematical and computational modelling

^** Corresponding author. Email: wetton{at}math.ubc.ca^*** Email: Peter.Berg{at}uoit.ca^**** Email: caglara{at}uwgb.edu^***** Email: kpromisl{at}math.msu.edu^****** Email: jean.st-pierre{at}ballard.com A mathematical model describing the effects of electrical couplingof proton exchange membrane unit fuel cells through shared bipolarplates is developed. Here, the unit cells are described by simple,steady-state, 1D models appropriate for straight reactant gaschannel designs. A linear asymptotic version of the model isused to give analytic insight into the effect of the coupling,including estimates of the extent of the coupling in terms ofthe number of adjacent cells affected. An efficient numericalmethod is developed to solve the non-linear coupled system.Numerical results showing the effects on stack voltage due toa single cell with anomalous oxidant flow rate are given. Theeffects on stack performance due to end plate effects are alsogiven. It is shown that electrical coupling has a significanteffect on fuel cell performance.     

On the q -Convolution on the Line

The investigation of a q -analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, with special attention to the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras (with respect to the q -convolution), depending on a parameter s>0 , is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and is shown to be the quotient of an algebra studied in a previous paper modulo the kernel of a q -analogue of the Fourier transform. This result has an analytic interpretation in terms of analytic functions, whose q -moments have a (fast) decreasing behavior and allows the extension of Koornwinder's inversion formula for the q -Fourier transform. A few results on the invertibility of functions with respect to the q -convolution are also obtained and they are applied to the solution of certain simple linear q -difference equations with polynomial coefficients.     

On the q -Convolution on the Line

The investigation of a q -analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, with special attention to the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras (with respect to the q -convolution), depending on a parameter s>0 , is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and is shown to be the quotient of an algebra studied in a previous paper modulo the kernel of a q -analogue of the Fourier transform. This result has an analytic interpretation in terms of analytic functions, whose q -moments have a (fast) decreasing behavior and allows the extension of Koornwinder's inversion formula for the q -Fourier transform. A few results on the invertibility of functions with respect to the q -convolution are also obtained and they are applied to the solution of certain simple linear q -difference equations with polynomial coefficients.     

Best rational function approximation to Laplace transform inversion using a window function

The best rational function approximation for Laplace transform inversion due to Longman is modified by the introduction of an appropriate “window” function. This window function enables one to approximate the inverse transform f(t) by a linear combination g_n(t) of n exponential functions accurately in a given interval about a given point along the t-axis. It is proved that the sequence of approximants {g_n(t)}_{n = 1}^∞ converges to f(t) in the mean. The method is illustrated by some numerical examples.     

Harmonic functions and volume growth of fingerless ends

Let M be a noncompact complete Riemannian manifold with finitely many ends. In this paper we study the existence of Green's function for the p-Laplace equation on M in terms of a certain volume growth. We also show that the dimension of the linear space of polynomial growth harmonic functions is finite if a volume comparison condition and a mean value inequality for nonnegative subharmonic functions hold in sufficiently large parts of each end. Received June 9, 1999 / Published online July 3, 2000     

Sensitivity and Approximation of Coupled Fluid-Structure Equations by Virtual Control Method

The formulation of a particular fluid--structure interaction as an optimal control problem is the departure point of this work. The control is the vertical component of the force acting on the interface and the observation is the vertical component of the velocity of the fluid on the interface. This approach permits us to solve the coupled fluid--structure problem by partitioned procedures. The analytic expression for the gradient of the cost function is obtained in order to devise accurate numerical methods for the minimization problem. Numerical results arising from blood flow in arteries are presented. To solve the optimal control problem numerically, we use a quasi-Newton method which employs the analytic gradient of the cost function and the approximation of the inverse Hessian is updated by the Broyden, Fletcher, Goldforb, Shano (BFGS) scheme. This algorithm is faster than fixed point with relaxation or block Newton methods.     

Analytic Feynman integrals of transforms of variation of cylinder type functions over Wiener paths in abstract Wiener space

In this paper, we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of cylinder type functions defined over Wiener paths in abstract Wiener space. We also derive the analytic Feynman integral of the conditional Fourier-Feynman transform for the product of the cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors.     

具有缺项系数的几类解析函数族的性质

S*表示所有在单位圆盘 D 内解析且满足条件 f(0)=f′ (0)-1=0的星形函数族, K 表示所有在 D内解析且满足条件 f(0)=f′ (0)-1=0 的凸函数族, P 表示所有在 D 内解析且满足条件p(0)=1, Rep(z)>0 的函数族. 设P_n={p(z): p(z)=1+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ P}, S*_n={f (z): f(z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ S*}, K_n={f (z): f (z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ K}. L_Sn*={g(z)=ln f(z)/z, f ∈ S_n*}, 其中对数函数取使得ln1=0的那个单值解析分支. 该文研究了函数族S_n^*, K_n和L_Sn*的性质, 找出了解析函数族L_Sn*的极值点与支撑点,并对S*_n与K_n的极值点和支撑点作了一些探讨.     

Complex Angular Momentum Diagonalization of the Bethe-Salpeter Structure in General Quantum Field Theory

The Complex Angular Momentum (CAM) representation of (scalar) fourpoint functions has been previously established starting from the general principles of local relativistic Quantum Field Theory (QFT). Here, we carry out the diagonalization of the general t-channel Bethe-Salpeter (BS) structure of four-point functions in the corresponding CAM variable 5^t, for all negative values of the squared-energy variable t. This diagonalization is closely related to the existence of BS-equations for the absorptive parts in the crossed channels, interpreted as convolution equations with spectral properties. The production of Regge poles equipped with factorized residues involving Euclidean three-point functions appears as conceptually built-in in the analytic axiomatic framework of QFT. The existence of leading Reggeon terms governing the asymptotic behaviour of the four-point function at fixed t is strictly conditioned by the asymptotic behaviour of a g lobal Bethe-Salpeter kernel of the theory.     

Least squares coefficients for a quadrature formula for Laplace transform inversion

A linear iterative method of least squares approximation of functions by exponentials due to Miller [9] is adapted to derive a set of least squares coefficients for an approximate Laplace transform inversion formula eq. (1). An earlier assumption made by Zakian [2] - that the approximation to the Laplace transform inverse will improve provided the approximation to the Dirac delta function is improved - is shown to be not substantiated for a number of test functions.