Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic...Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic-trigonometric spline space in order to resolve the theo- retical problem that there is not an explicit orthogonai basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.展开更多
Spline wavelet and orthogonal wavelet are two widely used wavelet methods. In this paper, comparison of these two methods has been made, including their algorithm, properties and results of signal processing in analyt...Spline wavelet and orthogonal wavelet are two widely used wavelet methods. In this paper, comparison of these two methods has been made, including their algorithm, properties and results of signal processing in analytical chemistry signals. It is found that spline wavelet is more effective than orthogonal wavelet in processing high noise signals. The curves obtained from spline wavelet are closer to the theoretical ones than those obtained from orthogonal wavelet and the errors of spline wavelet are smaller than those of orthogonal wavelet.展开更多
In this paper,we present the use of the orthogonal spline collocation method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations.This method uses the Hermite basis fu...In this paper,we present the use of the orthogonal spline collocation method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations.This method uses the Hermite basis functions,by which physical quantities are approximatedwith their values and derivatives associatedwith Gaussian points.The convergence rate with order O(h4+t2)and the stability of the scheme are proved.Conservation properties are shown in both theory and practice.Extensive numerical experiments are presented to validate the numerical study under consideration.展开更多
We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the u...We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval.Using an extension of the analysis of Douglas and Dupont[23]for Dirichlet boundary conditions,we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space.We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear systems.We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.展开更多
The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles.When the ADI technique is coupled with orthogona...The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles.When the ADI technique is coupled with orthogonal spline collocation(OSC)for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order.In[2],we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms.A natural question that arises is:Does this method have an extension to non-rectangular regions?In this paper,we present a simple idea of how the ADI OSC technique can be extended to some such regions.Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem(TPBVP).We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.展开更多
A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Ve...A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Verlag;(1992),pp.57–76].In this work,we first modify these wavelets to solve the elliptic(partially)Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods.We generalize this construction to two dimensional case by Tensor product space.In addition,the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed.We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant.Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel,it is solved by multiple knot B-spline wavelet method that yields a very well approximation.Finally,some numerical examples are given to support our theoretical results.展开更多
基金Supported by the National Natural Science Foundation of China(60933008,61272300 and 11226327)the Science&Technology Program of Shanghai Maritime University(20120099)
文摘Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic-trigonometric spline space in order to resolve the theo- retical problem that there is not an explicit orthogonai basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.
基金Project supported by the National Natural Science Foundation of China (No. 29675033)Natural Science Foundation of Guangdong Province (No. 960006)
文摘Spline wavelet and orthogonal wavelet are two widely used wavelet methods. In this paper, comparison of these two methods has been made, including their algorithm, properties and results of signal processing in analytical chemistry signals. It is found that spline wavelet is more effective than orthogonal wavelet in processing high noise signals. The curves obtained from spline wavelet are closer to the theoretical ones than those obtained from orthogonal wavelet and the errors of spline wavelet are smaller than those of orthogonal wavelet.
文摘In this paper,we present the use of the orthogonal spline collocation method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations.This method uses the Hermite basis functions,by which physical quantities are approximatedwith their values and derivatives associatedwith Gaussian points.The convergence rate with order O(h4+t2)and the stability of the scheme are proved.Conservation properties are shown in both theory and practice.Extensive numerical experiments are presented to validate the numerical study under consideration.
基金The research of J.C.Lopez-Marcos is supported in part by Ministerio de Ciencia e Innovacion,MTM2011-25238.
文摘We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval.Using an extension of the analysis of Douglas and Dupont[23]for Dirichlet boundary conditions,we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space.We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear systems.We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.
基金This work was supported by grant no.13328 from the Petroleum Institute,Abu Dhabi,UAE.
文摘The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles.When the ADI technique is coupled with orthogonal spline collocation(OSC)for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order.In[2],we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms.A natural question that arises is:Does this method have an extension to non-rectangular regions?In this paper,we present a simple idea of how the ADI OSC technique can be extended to some such regions.Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem(TPBVP).We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.
文摘A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Verlag;(1992),pp.57–76].In this work,we first modify these wavelets to solve the elliptic(partially)Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods.We generalize this construction to two dimensional case by Tensor product space.In addition,the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed.We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant.Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel,it is solved by multiple knot B-spline wavelet method that yields a very well approximation.Finally,some numerical examples are given to support our theoretical results.
