Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables

Under some conditions of uniform integrability and appropriate conditions, mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables are obtained. Our results extend and improve the results of [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008) 289-300] and [M. Ordóñez Cabrera, A. Volodin, Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability, J. Math. Anal. Appl. 305 (2005) 644-658].     

Exact solutions of nonlinear dispersive K(m, n) model with variable coefficients

The variable-coefficient Korteweg-de Vries (KdV) equation with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condense was presented to describe the dynamics of nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. To understand the role of nonlinear dispersion in this variable-coefficient model, we introduce and study a new variable-coefficient KdV with nonlinear dispersion (called vc-K(m, n) equation). With the aid of symbolic computation, we obtain its compacton-like solutions and solitary pattern-like solutions. Moreover, we also present some conservation laws for both vc-K⁺(n, n) equation and vc-K⁻(n, n) equation.     

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface Sg of genus g grows asymptotically like

c(g)n5(g−1)/2−1γnn!     

Convergence of an adaptive Ka?anov FEM for quasi-linear problems

We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Ka?anov iteration and a mesh adaptation step is performed after each linear solve. The method is thus inexact because we do not solve the discrete nonlinear problems exactly, but rather perform one iteration of a fixed point method (Ka?anov), using the approximation of the previous mesh as an initial guess. The convergence of the method is proved for any reasonable marking strategy and starting from any initial mesh. We conclude with some numerical experiments that illustrate the theory.     

Causality analysis of frequency‐dependent wave attenuation

The work is inspired by thermo‐ and photoacoustic imaging, where recent efforts are devoted to take into account attenuation and varying wave speed parameters. In this paper, we derive and analyze causal equations describing the propagation of attenuated pressure waves. We also review standard models, like frequency power laws, and the thermo‐viscous equation and show that they lack causality in the parameter range relevant for biological photoacoustic imaging. To discuss causality in mathematical rigor we use the results and concepts of linear system theory. We present some numerical experiments, which show the physically unmeaningful behavior of standard attenuation models, and the realistic behavior of the novel models. Copyright © 2010 John Wiley & Sons, Ltd.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES

We give a brief discussion of some of the contributions of Peter Lax to Computational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We develop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax-Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh’s infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.     

A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $ \mathcal{O}( h^6 + \tau^2) $ in the discrete $ L_2 $ norm with mesh size $ h $ and the time step $ \tau $. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.     

Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

We consider the development of implicit‐explicit time integration schemes for optimal control problems governed by the Goldstein–Taylor model. In the diffusive scaling, this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman–Enskog‐type limiting procedure. For the class of stiffly accurate implicit–explicit Runge–Kutta methods, the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1770–1784, 2014