Rodrigues solutions of the dirac equation for shape-invariant potentials: Master function approach

We show that a Schrödinger-like differential equation for the upper spinor component, derived from the Dirac equation for a charged spinor in spherically symmetric electromagnetic potentials, can be transformed into the Schrödinger equation with some shape-invariant potentials. By choosing different electrostatic potentials and relativistic energies and also introducing new functions and changing the variables, we show that this equation transforms into the differential equations in mathematical physics. We solve these equations using the master function approach and write the spinor wave functions in terms of Rodrigues polynomials associated with these differential equations.     

Pricing American options when asset prices jump

We present a transformation that helps price American options on assets that are modeled by a diffusion as well as a jump component. The presence of a jump component circumvents some shortcomings of the Black-Scholes model. The proposed transformation essentially transforms the arising free-boundary partial integro-differential equation (PIDE) into a sequence of fixed-boundary PIDEs which are much easier to solve. Finally, we provide numerical results illustrating convergence of the scheme and comparisons to other methods.     

具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式

构造了一种解具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式．基本思想是离散Landau-Lifshitz方程成常微分方程组,应用李群方法和显式Runge-Kutta方法解常微分方程组．数值试验比较了两方法的保平方守恒特性和精度,得出李群方法(RK-Cayley方法)比相应的Runge-Kutta(RK)方法有更好的精度和保平方守恒特性．     

Polynomial decay and control of a 1−d hyperbolic-parabolic coupled system

In this paper we consider a linearized model for fluid-structure interaction in one space dimension. The domain where the system evolves consists in two parts in which the wave and heat equations evolve, respectively, with transmission conditions at the interface. First of all we develop a careful spectral asymptotic analysis on high frequencies for the underlying semigroup. It is shown that the semigroup governed by the system can be split into a parabolic and a hyperbolic projection. The dissipative mechanism of the system in the domain where the heat equation holds produces a slow decay of the hyperbolic component of solutions. According to this analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Next, we discuss the problem of null-controllability of the system when the control acts on the boundary of the domain where the heat equation holds. The key observability inequality of the dual system with observation on the heat component is derived though a new Ingham-type inequality, which in turn, thanks to our spectral analysis, is a consequence of a known observability inequality of the same system but with observation on the wave component.     

A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations. We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.     

Approximation of Solution Components for Ill-Posed Problems by the Tikhonov Method with Total Variation

An ill-posed problem in the form of a linear operator equation given on a pair of Banach spaces is considered. Its solution is representable as a sum of a smooth and a discontinuous component. A stable approximation of the solution is obtained using a modified Tikhonov method in which the stabilizer is constructed as a sum of the Lebesgue norm and total variation. Each of the functionals involved in the stabilizer depends only on one component and takes into account its properties. Theorems on the componentwise convergence of the regularization method are stated, and a general scheme for the finite-difference approximation of the regularized family of approximate solutions is substantiated in the n-dimensional case.     

Zur Kontinuum-Magneto-Gasdynamik Stationäre rotationssymmetrische Strömung eines vollkommen leitenden Plasmas

Summary In the case of rotational symmetry for the steady motion of a perfectly conducting plasma a differential equation of 2nd order for the stream function is given by completingBernoulli's equation, the equation for the moment of rotation and the equation for the vorticity respectively by the azimutal component of the magnetic field and the azimutal component of electric current density.

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Adolf Busemann zum 60. Geburtstag     

Different Definitions of Algebraically Closed Skew Fields

We consider an algebraically closed (in the sense of solvability of arbitrary polynomial equations) skew field constructed by Makar-Limanov. It is shown that every generalized polynomial equation with more than one homogeneous component has a non-zero solution. We also look into P. Cohn's approach to defining algebraically closed non-commutative skew fields and treat some related problems.     

Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.     

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·     

基于二次多项式的积极集光滑化max函数及其在无约束minimax问题中的应用

本文基于分段二次多项式方程,构造了一种积极集策略的光滑化max函数.通过给出与光滑化max函数相关的分量函数指标集的直接计算方法,将分段二次多项式方程转化为一般二次多项式方程.利用二次多项式方程根的性质,给出了该光滑化max函数的稳定计算策略,证明了其具有一阶光滑性,其梯度函数具有局部Lipschitz连续性和强半光滑性.该光滑化max函数仅与函数值较大的分量函数相关,适用于含分量函数较多且复杂的max函数的问题.为了验证其效率,本文基于该函数构造了一种解含多个复杂分量函数的无约束minimax问题的光滑化算法,数值实验表明了该光滑化max函数的可行性及有效性.     

广义缓坡方程

运用表面波Hamilton方法和缓坡逼近假定,分析缓变三维流场和非平整海底对波浪传播的影响,推导出广义缓坡方程。海底地形由两个分量组成:慢变分量,其水平长度尺度大于表面波的波长;快变分量,其振幅与表面波的波长相比为一小量,但是其频率与表面波频率相当。该广义缓坡方程具有广泛的适用范围,以下著名的缓坡方程成为它的特例:经典的Berkhoff缓坡方程;包含环境流效应的Kirby缓坡方程;描述波状海底特征的Dingemans缓坡方程。另外,同时也得到了描述环境流场和快变海底效应的广义浅水方程。     

A new a posteriori error estimate for the Morley element

In this paper, we present a posteriori error analysis for the nonconforming Morley element of the fourth order elliptic equation. We propose a new residual-based a posteriori error estimator and prove its reliability and efficiency. These results refine those of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) by dropping two edge jump terms in both the energy norm of the error and the estimator, and those of Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) by showing the efficiency in the sense of Verfürth (A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, New York, 1996). Moreover, the normal component in the estimators of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) and Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) is dropped, and therefore only the tangential component of the stress on each edge comes into the estimator. In addition, we generalize these results to three dimensional case.     

On a structural acoustic model which incorporates shear and thermal effects in the structural component

In this paper we consider a structural acoustic model which takes account of thermal effects over and above displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the structural medium is a Reissner-Mindlin plate into which an additional degree of freedom, viz. temperature variation in the plate, has been introduced and the constitutive equations for the structural acoustic model couple parabolic dynamics with hyperbolic dynamics. We show unique solvability of the mathematical model and investigate the effect of the presence of thermal effects on the mechanical dissipation devices needed to attain uniform stabilization of the two-dimensional model in which the structural component is a Timoshenko beam. It turns out that, as in linear structural acoustic models which use the Euler-Bernoulli equation or the Kirchoff equation to describe the deflections of the thermo-elastic structural medium, uniform stabilization of the energy associated with the model can be attained without introducing mechanical dissipation at the free edge of the beam. Open problems with regard to the stabilization of the three-dimensional model are outlined.     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

Solution of the advection-dispersion equation with free exit boundary

I investigated the exit boundary condition for the advection-dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non-zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite-domain solution.     

Thermoactivation theory of wear in plastics

It has been shown that wear, being a thermofluctuation process, can be described by an equation analogous to Zhurkov's law of simple fracture. In fatigue wear the maximum value of the activation energy u₀ is close to the u for the rupture of chemical bonds. Abrasive wear is an almost barrierless process, therefore in the case of mixed wear the presence of an abrasive component sharply reduced u₀. The thermoactivation equation describes the effect of temperature on wear only within the limits of one physical state of the polymer. In transition zones it is necessary to take into account the role of the sharp change in elongation at break, which introduces extrema into the temperature wear curve.Mekhanika Polimerov, Vol. 2, No. 6, pp. 867–874, 1966     

The polyharmonic local sine transform: A new tool for local image analysis and synthesis without edge effect

We introduce a new local sine transform that can completely localize image information both in the space domain and in the spatial frequency domain. This transform, which we shall call the polyharmonic local sine transform (PHLST), first segments an image into local pieces using the characteristic functions, then decomposes each piece into two components: the polyharmonic component and the residual. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the so-called polyharmonic equation (e.g., Laplace's equation, biharmonic equation, etc.) given the boundary values (the pixel values along the boundary created by the characteristic function). Subsequently this component is subtracted from the original local piece to obtain the residual. Since the boundary values of the residual vanish, its Fourier sine series expansion has quickly decaying coefficients. Consequently, PHLST can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. Combining this ability with the quickly decaying coefficients of the residuals, PHLST is also effective for image approximation, which we demonstrate using both synthetic and real images. In addition, we introduce the polyharmonic local Fourier transform (PHLFT) by replacing the Fourier sine series above by the complex Fourier series. With a slight sacrifice of the decay rate of the expansion coefficients, PHLFT allows one to compute local Fourier magnitudes and phases without revealing the edge effect (or Gibbs phenomenon), yet is invertible and useful for various filtering, analysis, and approximation purposes.     

Properties of solutions of two second-order differential equations with the Painlevé property

Theoretical and Mathematical Physics - We consider a Hamiltonian system equivalent to the Painlevé II equation with respect to one component and to the Painlevé XXXIV equation with...