It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollab...It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.展开更多
General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation a...General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation and barycentric rational Her- mite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.展开更多
This paper presents a distributed planar leader-follower formation maneuver control strategy for multi-agent systems with different agent dynamic models.This method is based on the barycentric coordinate-based(BCB)con...This paper presents a distributed planar leader-follower formation maneuver control strategy for multi-agent systems with different agent dynamic models.This method is based on the barycentric coordinate-based(BCB)control,which can be performed in the local coordinate frame of each agent with required local measurements.By exploring the properties of BCB Laplacians,a time-varying target formation can be BCB localizable by a sufficient number of leaders uniquely,and this formation is converted from a given nominal formation with geometrical similarity transformation.The proposed control laws can continuously maneuver collective single-and double-integrator agents to achieve a translation,scale,rotation,or even their compositions in various directions.For the formation shape control problem of multi-car systems with/without saturation constraints,the obtained control performance can preserve good robustness.Global stability is also proven by mathematical derivations and verified by numerical simulations.展开更多
This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by...This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.展开更多
In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with cor...In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with corresponding weight functions. Any partial interpolant is a piecewise cubics defined on a split of the triangle, while the weight function is just the respective barycentric coordinate. Hence the interpolant can be regarded as a piecewise quartic. We device a simple algorithm for the evaluation of the interpolant. It's easy to represent the interpolant with B-net method. We also depict the Franke's function and its interpolant, the illustration of which shows good visual effect of the scheme.展开更多
Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough hi...Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.展开更多
In a connected graph G, the distance d（u, v） denotes the distance between two vertices u and v of G. Let W = {w1, w2,……, wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r（v1W...In a connected graph G, the distance d（u, v） denotes the distance between two vertices u and v of G. Let W = {w1, w2,……, wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r（v1W） of v with respect to W is the k-tuple （d（v, w1）, d（v, w2）,…, d（v, wk））. The set W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by β（G）. Metric dimension is a generalization of affine dimension to arbitrary metric spaces （provided a resolving set exists）. In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs Cay （Zn Z2）. We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of barycentric subdivision of Cayley graphs Cay （Zn Z2）.展开更多
This paper presents a result of approximation an arc circles by using a quartic Bezier curve. Based on the barycentric coordinates of two and three combination of control points, the interior control points are determ...This paper presents a result of approximation an arc circles by using a quartic Bezier curve. Based on the barycentric coordinates of two and three combination of control points, the interior control points are determined by forcing the curvature at median point as similar as the given curvature at end points. Hausdorff distance is used to investigate the order of accuracy compare to the actual arc circles through central angle of . We found that the optimal approximation order is eight which is somewhat similar to preceding methods in the literatures.展开更多
The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rat...The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.展开更多
Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics.The numerical error resulting from the stress determination is a m...Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics.The numerical error resulting from the stress determination is a main concern as it is always a function of parameters in discretization.To improve the accuracy of stress calculation,a novel meshless barycentric rational interpolation collocation method(BRICM)is proposed.The derivatives of the shear stress on the calculation path are determined by using the differential matrix which converts the differential form of the equations of equilibrium into a series of algebraic equations.The advantage of the proposed method is that the auxiliary lines,grids,and error accumulation which are commonly used in traditional shear difference methods(SDMs)are not required.Simulation and experimental results indicate that the proposed meshless method is able to provide high computational accuracy in the full-field stress determination.展开更多
Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fa...Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fast capabilities for Clenshaw-Curtis and also Gauss-Legendre,-Jacobi,-Hermite,and-Laguerre quadrature,based on algorithms of Waldvogel and Glaser,Liu and Rokhlin.Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles,fractional derivatives and integrals,functions defined on unbounded intervals,and the fast computation of weights for barycentric interpolation.展开更多
In last time,the series of virtual internal bond model was proposed for solving rock mechanics problems.In these models,the rock continuum is considered as a structure of discrete particles connected by normal and she...In last time,the series of virtual internal bond model was proposed for solving rock mechanics problems.In these models,the rock continuum is considered as a structure of discrete particles connected by normal and shear springs(bonds).It is well announced that the normal springs structure corresponds to a linear elastic solid with a fixed Poisson ratio,namely,0.25 for threedimensional cases.So the shear springs used to represent the diversity of the Poisson ratio.However,the shearing force calculation is not rotationally invariant and it produce difficulties in application of these models for rock mechanics problems with sufficient displacements.In this letter,we proposed the approach to support the diversity of the Poisson ratio that based on usage of deformable Voronoi cells as set of particles.The edges of dual Delaunay tetrahedralization are considered as structure of normal springs(bonds).The movements of particle’s centers lead to deformation of tetrahedrals and as result to deformation of Voronoi cells.For each bond,there are the corresponded dual face of some Voronoi cell.We can consider the normal bond as some beam and in this case,the appropriate face of Voronoi cell will be a cross section of this beam.If during deformation the Voronoi face was expand,then,according Poisson effect,the length of bond should be decrees.The above mechanism was numerically investigated and we shown that it is acceptable for simulation of elastic behavior in 0.1–0.3 interval of Poisson ratio.Unexpected surprise is that proposed approach give possibility to simulate auxetic materials with negative Poisson’s ratio in interval from–0.5 to–0.1.展开更多
In this paper,we propose an efficient computational method for converting local coordinates to world coordinates using specially structured coordinate data.The problem in question is the computation of world coordinat...In this paper,we propose an efficient computational method for converting local coordinates to world coordinates using specially structured coordinate data.The problem in question is the computation of world coordinates of an object throughout a motion,assuming that we only know the changing coordinates of some fixed surrounding reference points in the local coordinate system of the object.The proposed method is based on barycentric coordinates;by taking the aforementioned static positions as the vertices of a polyhedron,we can specify the coordinates of the object in each step with the help of barycentric coordinates.This approach can significantly help us to achieve more accurate results than by using other possible methods.In the paper,we describe the problem and barycentric coordinate-based solution in detail.We then compare the barycentric method with a technique based on transformation matrices,which we also tested for solving our problem.We also present various diagrams that demonstrate the efficiency of our proposed approach in terms of precision and performance.展开更多
In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,tw...In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,two stable discretizations for the Stokes equations are developed,which can be viewed as the extensions of the P2-P0 and the Q2-(discontinuous)P1 elements,respectively,to polygonal meshes.Numerical results are presented,which support our theoretical claims.展开更多
This work introduces a novel tool for interactive, real-time affine transformations of two dimensional IFS fractals. The tool uses some of the nice properties of the barycentric coordinates that are assigned to the po...This work introduces a novel tool for interactive, real-time affine transformations of two dimensional IFS fractals. The tool uses some of the nice properties of the barycentric coordinates that are assigned to the points that constitute the image ofa fractal, and thus enables any affine transformation of the affine basis, done by click-and-drag, to be immediately followed by the same affine transformation of the fractal. The barycentric coordinates can be relative to an arbitrary affine basis of ~2, but in order to have a better control over the fractal, a kind of minimal simplex that contains the fractal attractor is used.展开更多
In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is appli...In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.展开更多
文摘It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.
基金supported by the grant of Key Scientific Research Foundation of Education Department of Anhui Province, No. KJ2014A210
文摘General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation and barycentric rational Her- mite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.
基金This work was supported by National Natural Science Foundation of China(Grant No.61673327)Industrial Development and Foster Project of Yangtze River Delta Research Institute of NPU,Taicang(Grant No.CY20210202).
文摘This paper presents a distributed planar leader-follower formation maneuver control strategy for multi-agent systems with different agent dynamic models.This method is based on the barycentric coordinate-based(BCB)control,which can be performed in the local coordinate frame of each agent with required local measurements.By exploring the properties of BCB Laplacians,a time-varying target formation can be BCB localizable by a sufficient number of leaders uniquely,and this formation is converted from a given nominal formation with geometrical similarity transformation.The proposed control laws can continuously maneuver collective single-and double-integrator agents to achieve a translation,scale,rotation,or even their compositions in various directions.For the formation shape control problem of multi-car systems with/without saturation constraints,the obtained control performance can preserve good robustness.Global stability is also proven by mathematical derivations and verified by numerical simulations.
文摘This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.
文摘In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with corresponding weight functions. Any partial interpolant is a piecewise cubics defined on a split of the triangle, while the weight function is just the respective barycentric coordinate. Hence the interpolant can be regarded as a piecewise quartic. We device a simple algorithm for the evaluation of the interpolant. It's easy to represent the interpolant with B-net method. We also depict the Franke's function and its interpolant, the illustration of which shows good visual effect of the scheme.
文摘Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.
基金Supported by the National University of Sciences and Technology(NUST),H-12,Islamabad,Pakistan
文摘In a connected graph G, the distance d（u, v） denotes the distance between two vertices u and v of G. Let W = {w1, w2,……, wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r（v1W） of v with respect to W is the k-tuple （d（v, w1）, d（v, w2）,…, d（v, wk））. The set W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by β（G）. Metric dimension is a generalization of affine dimension to arbitrary metric spaces （provided a resolving set exists）. In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs Cay （Zn Z2）. We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of barycentric subdivision of Cayley graphs Cay （Zn Z2）.
文摘This paper presents a result of approximation an arc circles by using a quartic Bezier curve. Based on the barycentric coordinates of two and three combination of control points, the interior control points are determined by forcing the curvature at median point as similar as the given curvature at end points. Hausdorff distance is used to investigate the order of accuracy compare to the actual arc circles through central angle of . We found that the optimal approximation order is eight which is somewhat similar to preceding methods in the literatures.
基金the National Natural Science Foundation of China (Nos. 11472041,11532002,11772049,and 11802320)。
文摘The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.
基金Project supported by the National Key R&D Program of China(No.2018YFF01014200)the National Natural Science Foundation of China(Nos.11727804,11872240,12072184,12002197,and 51732008)the China Postdoctoral Science Foundation(Nos.2020M671070 and 2021M692025)。
文摘Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics.The numerical error resulting from the stress determination is a main concern as it is always a function of parameters in discretization.To improve the accuracy of stress calculation,a novel meshless barycentric rational interpolation collocation method(BRICM)is proposed.The derivatives of the shear stress on the calculation path are determined by using the differential matrix which converts the differential form of the equations of equilibrium into a series of algebraic equations.The advantage of the proposed method is that the auxiliary lines,grids,and error accumulation which are commonly used in traditional shear difference methods(SDMs)are not required.Simulation and experimental results indicate that the proposed meshless method is able to provide high computational accuracy in the full-field stress determination.
基金supported by the MathWorks,Inc.,King Abdullah University of Science and Technology (KAUST) (Award No. KUK-C1-013-04)the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC (Grant Agreement No. 291068)
文摘Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fast capabilities for Clenshaw-Curtis and also Gauss-Legendre,-Jacobi,-Hermite,and-Laguerre quadrature,based on algorithms of Waldvogel and Glaser,Liu and Rokhlin.Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles,fractional derivatives and integrals,functions defined on unbounded intervals,and the fast computation of weights for barycentric interpolation.
文摘In last time,the series of virtual internal bond model was proposed for solving rock mechanics problems.In these models,the rock continuum is considered as a structure of discrete particles connected by normal and shear springs(bonds).It is well announced that the normal springs structure corresponds to a linear elastic solid with a fixed Poisson ratio,namely,0.25 for threedimensional cases.So the shear springs used to represent the diversity of the Poisson ratio.However,the shearing force calculation is not rotationally invariant and it produce difficulties in application of these models for rock mechanics problems with sufficient displacements.In this letter,we proposed the approach to support the diversity of the Poisson ratio that based on usage of deformable Voronoi cells as set of particles.The edges of dual Delaunay tetrahedralization are considered as structure of normal springs(bonds).The movements of particle’s centers lead to deformation of tetrahedrals and as result to deformation of Voronoi cells.For each bond,there are the corresponded dual face of some Voronoi cell.We can consider the normal bond as some beam and in this case,the appropriate face of Voronoi cell will be a cross section of this beam.If during deformation the Voronoi face was expand,then,according Poisson effect,the length of bond should be decrees.The above mechanism was numerically investigated and we shown that it is acceptable for simulation of elastic behavior in 0.1–0.3 interval of Poisson ratio.Unexpected surprise is that proposed approach give possibility to simulate auxetic materials with negative Poisson’s ratio in interval from–0.5 to–0.1.
基金supported by the construction EFOP-3.6.3-VEKOP-16-2017-00002supported by the European Union,co-financed by the European Social Fund.
文摘In this paper,we propose an efficient computational method for converting local coordinates to world coordinates using specially structured coordinate data.The problem in question is the computation of world coordinates of an object throughout a motion,assuming that we only know the changing coordinates of some fixed surrounding reference points in the local coordinate system of the object.The proposed method is based on barycentric coordinates;by taking the aforementioned static positions as the vertices of a polyhedron,we can specify the coordinates of the object in each step with the help of barycentric coordinates.This approach can significantly help us to achieve more accurate results than by using other possible methods.In the paper,we describe the problem and barycentric coordinate-based solution in detail.We then compare the barycentric method with a technique based on transformation matrices,which we also tested for solving our problem.We also present various diagrams that demonstrate the efficiency of our proposed approach in terms of precision and performance.
基金supported by the NSFC grant 11671210 and 12171244.
文摘In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,two stable discretizations for the Stokes equations are developed,which can be viewed as the extensions of the P2-P0 and the Q2-(discontinuous)P1 elements,respectively,to polygonal meshes.Numerical results are presented,which support our theoretical claims.
文摘This work introduces a novel tool for interactive, real-time affine transformations of two dimensional IFS fractals. The tool uses some of the nice properties of the barycentric coordinates that are assigned to the points that constitute the image ofa fractal, and thus enables any affine transformation of the affine basis, done by click-and-drag, to be immediately followed by the same affine transformation of the fractal. The barycentric coordinates can be relative to an arbitrary affine basis of ~2, but in order to have a better control over the fractal, a kind of minimal simplex that contains the fractal attractor is used.
基金partially supported by National Natural Science Foundation of China(11772165,11961054,11902170)Key Research and Development Program of Ningxia(2018BEE03007)+1 种基金National Natural Science Foundation of Ningxia(2018AAC02003,2020AAC03059)Major Innovation Projects for Building First-class Universities in China’s Western Region(Grant No.ZKZD2017009).
文摘In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.
