A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

Generator,multiquadric generator,quasi-interpolation and multiquadric quasi-interpolation

The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.

A meshless scheme for partial differential equations based on multiquadric trigonometric B-spline quasi-interpolation

Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This scheme takes into ac- count the periodicity of the analytic solution by using derivatives of a periodic quasi-interpolant （multiquadric trigonometric B-spline quasi-interpolant） to approximate the spatial derivatives of the equations. Thus, it overcomes the difficulties of the previous schemes based on quasi-interpolation （requiring some additional boundary conditions and yielding unwanted high-order discontinuous points at the boundaries in the spatial domain）. Moreover, the scheme also overcomes the dif- ficulty of the meshless collocation methods （i.e., yielding a notorious ill-conditioned linear system of equations for large collocation points）. The numerical examples that are presented at the end of the paper show that the scheme provides excellent approximations to the analytic solutions.

QUASI-INTERPOLATION AND APPROXIMATION VIA NONSEPARABLE SCALING FUNCTION

Quasi-interpolation has been studied in many papers, e. g., [5]. Here we introduce nonseparable scaling function quasi-interpolation and show that its approximation can provide similar convergence properties as scalar wavelet system. Several equivalent statements of accuracy of nonseparable scaling function are also given. In the numerical experiments, it appears that nonseparable scaling function interpolation has better convergence results than scalar wavelet systems in some cases.

On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline Space S_(3)^(1,2)(∆_(mn)^((2)))

In this paper,based on the basis composed of two sets of splines with distinct local supports,cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation.The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points,which can reproduce any polynomial of nearly best degrees.And by means of the modulus of continuity,the estimation of the operator approximating a real sufficiently smooth function is reviewed as well.Moreover,the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation.And then the convergence results are worked out.

A MULTIVARIATE MULTIQUADRIC QUASI-INTERPOLATION WITH QUADRIC REPRODUCTION

In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can reproduce linear polynomials to the scheme quadric polynomials. Furthermore, we give the approximation error of the modified scheme. Our multivariate multiquadric quasi-interpolation scheme only requires information of lo- cation points but not that of the derivatives of approximated function. Finally, numerical experiments demonstrate that the approximation rate of our scheme is significantly im- proved which is consistent with the theoretical results.

Rational Quasi-Interpolation Approximation of Scattered Data in R^(3)

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree(n+1)to approximate the scattered data in R 3.We firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree(n+1).Then,based on the triangulation of the scattered nodes in R^(2),on each triangle a rational quasi-interpolation function is constructed.The constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree(n+1).By comparing accuracy,stability,and efficiency with the C^(1)-Tri-interpolation method of Goodman[16]and the MQ Shepard method,it is observed that our method has some computational advantages.

APPROXIMATE IMPLICITIZATION BASED ON RBF NETWORKS AND MQ QUASI-INTERPOLATION

In this paper, we propose a new approach to solve the approximate implicitization problem based on RBF networks and MQ quasi-interpolation. This approach possesses the advantages of shape preserving, better smoothness, good approximation behavior and relatively less data etc. Several numerical examples are provided to demonstrate the effectiveness and flexibility of the proposed method.

Applying Multiquadric Quasi-Interpolation to Solve KdV Equation

Quasi-interpolation is very useful in the study of approximation theory and its applications,since it can yield solutions directly without the need to solve any linear system of equations.Based on the good performance,Chen and Wu presented a kind of multiquadric （MQ） quasi-interpolation,which is generalized from the L D operator,and used it to solve hyperbolic conservation laws and Burgers’ equation.In this paper,a numerical scheme is presented based on Chen and Wu’s method for solving the Korteweg-de Vries （KdV） equation.The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative,and the forward divided difference to approximate the temporal derivative,where the spatial derivative is approximated by the derivative of the generalized L D quasi-interpolation operator.The algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.

SOME SHAPE-PRESERVING QUASI-INTERPOLANTS TO NON-UNIFORMLY DISTRIBUTED DATA BY MQ-B-SPLINES

Based on the definition of MQ-B-Splines,this article constructs five types of univariate quasi-interpolants to non-uniformly distributed data. The error estimates and the shape-preserving properties are shown in details.And examples are shown to demonstrate the capacity of the quasi-interpolants for curve representation.

A FAMILY OF BERNSTEIN QUASI-INTERPOLANTS ON[0,1]

Suppose that we want to approximate fC[0,1]by polynomials in P_n,using only its values on X_n={i/n,0≤i≤n}.This can be done by the Lagrange interpolant L_n f or the classical Bernstein polynomial B_n f.But,when n tends to infinity,L_n f does not converge to f in general and the convergence of B_n f to fis very slow.We define a family of operators B^(k)_n, n≥k,which are intermediate ones between B(0)_n=B^(1)_n=B_n and B^(n)_n=L_n,and we study some of their properties.In particular,we prove a Voronovskaja-type theorem which asserts that B^(k)_n f-f=0(n^(-[(k+2)/2))for f sufficiently regular. Moreover,B(k)_n f uses only values of B_n f and its derivaties and can be computed by De Casteljau or subdivision algorithms.

THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0,1]

In this paper, we will use the 2r-th Ditzian-Totik modulus of smoothness wψ2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn[2r-1]f for functions of the space Lp[0,1] (1≤ p≤ +∞).

L_p approximation by Bernstein-Kantorovich quasi-interpolants

Bernstein-Kantorovich quasi-interpolants K^（2r-1）n（f, x） are considered and direct, inverse and equivalence theorems with Ditzian-Totik modulus of smoothness ω^2rφ（f, t）p （1 ≤ p ≤＋∞） are obtained.

Weighted Approximation by Left Quasi-interpolants of Derivatives of Gamma Operators

In order to obtain much faster convergence, Miiller introduced the left Gamma quasi- interpolants and obtained an approximation equivalence theorem in terms of 2r wφ （f,t）p. Cuo extended the MiiUer＇s results to wφ^24 （f, t）∞. In this paper we improve the previous results and give a weighted approximation equivalence theorem.

POINTWISE SIMULTANEOUS APPROXIMATION BY LEFT GAMMA QUASI-INTERPOLANTS

Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence. We consider left Gamma quasi-interpolants and give a pointwise simultaneous approximation equivalence theorem with ωφλ^2r（f,t）∞ by means of unified the classical modulus and Ditzian-Totick modulus.

QUASI-INTERPOLATING OPERATORS AND THEIR APPLICATIONS IN HYPERSINGULAR INTEGRALS

The purpose of this paper is to propose and study a class of quasi-interpolating operators in multivariate spline space S-1/2(Delta(mn)(2*)) on non-uniform type-2 triangulation. Based on the operators, we construct cubature formula for two-dimensional hypersingular integrals. Some computing work have been done and the results are quite satisfactory.

Strong Converse Inequality for Left Gamma Quasi-Interpolants

Abstract The rate of convergence for the Gamma operators cannot be faster than $$O{\left( {\frac{1}{n}} \right)}$$. In order to obtain much faster convergence, quasi-interpolants in the sense of Sablonnière are considered. For the first time in the theory of quasi-interpolants, the strong converse inequality is solved in sup-norm with the K-functional $$K^{\alpha }_{\lambda } {\left( {f,t^{{2r}} } \right)}\;{\left( {0 \leqslant \lambda \leqslant 1,\;0

Weighted approximation by Bernstein quasiinterpolants for functions with singularities

We introduce a new type of modified Bernstein quasi-interpolants, which can be used to approximate functions with singularities. We establish direct, inverse, and equivalent theorems of the weighted approximation of this modified quasi-interpolants. Some classical results on approximation of continuous functions are generalized to the weighted approximation of functions with singularities.

A SPLINE METHOD FOR SOLVING TWO-DIMENSIONAL FREDHOLM INTEGRAL EQUATION OF SECOND KIND WITH THE HYPERSINGULAR KERNEL

Provides information on a study which proposed a spline method for solving two-dimensional Fredholm Integral Equations of second kind space with hypersingular kernels. Details on the quasi-interpolating operators; Information on the cubature formulas; Formulas of the approximation method.