SPLINE INTEGRAL OPERATORS EXACT ON POLYNOMIALS

We study integral spline operators of order k. exact on polynomials of degree 2m. with 0≤2m<k, having the form T_(k,t)^((m))f=∑ i∈J [∫_lf(x)C_(l,k)^(x)dx]N_IK, where {N_(l,k),i∈J} is the classical Bspline basis associated with the sequence t of knots on the interval I and C_(l,k)~is a linear combination of B-splines {N_(l+l,k),-m≤j≤m}. We prove a general theorem of eristence and uniqueness. Then we study the L^D -norms of these operators and error bounds for smooth furlctions f. We then obtain partial results about the L~∞--boundedness of T_(k,t)^((m)), independently of the pertition t. We also give the complete description of these operators in the case of a uniform partition of the real line.

Solving Two-Dimensional Moving-Boundary Problems with Meshless and Level Set Method

During the past decade, increasing attention has been given to the development of meshless methods using radial basis functions for the numerical solution of Partial Differential Equations （PDEs）. A level set method is a promising design tool for tracking, modelling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. In the conventional level set methods, the level set equation is solved to evolve the interface using a capturing Eulerian approach. The solving procedure requires an appropriate choice of the upwind schemes, reinitialization, etc. Our goal is to include Multiquadric Radial Basis Functions （MQ RBFs） into the level set method to construct a more efficient approach and stabilize the solution process with the adaptive greedy algorithm. This paper presents an alternative approach to the conventional level set methods for solving moving-boundary problems. The solution was compared to the solution calculated by the exact explicit lime integration scheme. The examples show that MQ RBFs and adaptive greedy algorithm is a very promising calculation scheme.

Symmetries,Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations

This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.

THE FLOW ANALYSIS OF FLUIDS IN FRACTAL RESERVOIR WITH THE FRACTIONAL DERIVATIVE

In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the flow models of fluids in fractal reservoirs with the fractional derivative. The flow characteristics of fluids through a fractal reservoir with the fractional order derivative are studied by using the finite integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of fluids flow through an infinite fractal reservoir is studied by using the Stehfest＇s inversion method of the numerical Laplace transform. It shows that the order of the fractional derivative affect the whole pressure behavior, particularly, the effect of pressure behavior of the early-time stage is larger The new type flow model of fluid in fractal reservoir with fractional derivative is provided a new mathematical model for studying the seepage mechanics of fluid in fractal porous media.

The Eulerian-Lagrangian Method with Accurate Numerical Integration

This paper is devoted to the study of the Eulerian-Lagrangian method(ELM)for convection-diffusion equations on unstructured grids with or without accurate numerical integration.We first propose an efficient and accurate algorithm to calculate the integrals in the Eulerian-Lagrangian method.Our approach is based on an algorithm for finding the intersection of two non-matching grids.It has optimal algorithmic complexity and runs fast enough to make time-dependent velocity fields feasible.The evaluation of the integrals leads to increased precision and the unconditional stability.We demonstrate by numerical examples that the ELM with our proposed algorithm for accurate numerical integration has the following two features:first it is much more accurate and more stable than the ones with traditional numerical integration techniques and secondly the overall cost of the proposed method is comparable with the traditional ones.

Lax Pairs and Integrability Conditions of Higher-Order Nonlinear Schrdinger Equations

We derive the Lax pairs and integrability conditions of the nonlinear Schrdinger equation with higher-order terms, complex potentials, and time-dependent coefficients. Cubic and quintic nonlinearities together with derivative terms are considered. The Lax pairs and integrability conditions for some of the well-known nonlinear Schrdinger equations, including a new equation which was not considered previously in the literature, are then derived as special cases. We show most clearly with a similarity transformation that the higher-order terms restrict the integrability to linear potential in contrast with quadratic potential for the standard nonlinear Schrdinger equation.