Numerical inversions of a source term in the FADE with a Dirichlet boundary condition using final observations

This paper deals with an inverse problem of determining a source term in the one-dimensional fractional advection-dispersion equation (FADE) with a Dirichlet boundary condition on a finite domain, using final observations. On the basis of the shifted Grünwald formula, a finite difference scheme for the forward problem of the FADE is given, by means of which the source magnitude depending upon the space variable is reconstructed numerically by applying an optimal perturbation regularization algorithm. Numerical inversions with noisy data are carried out for the unknowns taking three functional forms: polynomials, trigonometric functions and index functions. The reconstruction results show that the inversion algorithm is efficient for the inverse problem of determining source terms in a FADE, and the algorithm is also stable for additional data having random noises.     

Further results of recursive evaluation for compound distribution with the severity distribution of mixed type

Yang et al. [J.P. Yang, S.H. Cheng, Q. Wu, Recursive equations for compound distribution with the severity distribution of the mixed type, Science in China Series A 48 (2005) 594-609] investigated a recursive procedure for a kind of compound distributions with the number of claims belonging to (a,b)-family and the severity distribution of the mixed type. In this paper, we extend their results by assuming that the claim number belongs to a larger class. As applications, the excess-of-loss reinsurance treaty is discussed and concrete examples are considered in some detail.     

Dimensional precision simulation of the formed picture tube panel

In the forming process of picture tube panel, the accumulated residual stresses cause the formed part to shrink, and the thermal and mechanical loads cause the mold blocks to deform. These two factors result in large deviations on the dimensions of the formed panel, which are both modeled and simulated in this paper. For residual stresses analysis, a thermo-rheologically simple viscoelastic material model is introduced to consider the stresses relaxation effect and to describe the mechanical behavior according to the temperature change. The shrinkage of formed parts induced by the residual stresses is calculated based on the theory of shells, represented as an assembly of flat elements formed by combining the constant strain and the discrete Kirchhoff triangular elements. A thermoelastic model is presented to predict the deformation of the mold blocks during pressing, which is based on the steady mold temperature field and thermoelastic boundary element method. The integrated simulation results suggest the amounts that the mold cavity should be machined by, and have been verified by comparing the dimensional precision of the panels produced by the mold considering a uniform part shrinkage and mold expansion or the mold considering the predicted ununiform part shrinkage and mold deformation.     

A split least-squares characteristic mixed finite element method for Sobolev equations with convection term

In this paper, a split least-squares characteristic mixed finite element method for a kind of Sobolev equation with convection term is proposed, in which the characteristic method is based on the approximation of the material derivative term, that is, the time derivative term plus the convection term. The resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes and does not need to solve a coupled system of equations. Theory analysis shows that the method yields the approximate solutions with optimal accuracy in L²(Ω) norm for the primal unknown and in H(div;Ω) norm for the unknown flux, respectively. Numerical examples in one dimension, which are consistent with the theoretical results, are provided to demonstrate the characteristic behavior of this approach.     

Medial axis computation for planar free-form shapes

We present a simple, efficient, and stable method for computing—with any desired precision—the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.     

数列极限在实际中的应用

数列极限是一类重要的极限,在经济和数学领域中发挥着重要的作用。以举例的方式介绍数列极限在现实生活中的应用,包括几何中计算曲线所围图形的面积．求二次方程的数值解,购房按揭贷款分期还款问题,及连续复利问题。     

Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.     

SearchFill: A stochastic optimization code for detecting atomic vacancies in crystalline and non-crystalline systems

We present an implementation of a stochastic optimization algorithm applied to location of atomic vacancies. Our method labels an empty point in space as a vacancy site, if the total spatial overlap of a “virtual sphere”, centered around the point, with the surrounding atoms (and other vacancies) falls below a tolerance parameter. A Metropolis-like algorithm displaces the vacancies randomly, using an “overlap temperature” parameter to allow for acceptance of moves into regions with higher overlap, thus avoiding local minima. Once the algorithm has targeted a point with low overlap, the overlap temperature is decreased, and the method works as a steepest descent optimization.Our method, with only two free parameters, is able to detect the correct number and coordinates of vacancies in a wide spectrum of condensed-matter systems, from crystals to amorphous solids, in fact in any given set of atomic coordinates, without any need of comparison with a reference initial structure.