The exponential cubic B-spline algorithm for Fisher equation

In this study, exponential B-spline collocation method is set up for solving Fisher’s equation. Integration of Fisher’s equation is managed by use of the exponential cubic B-spline in space and the Crank–Nicolson method in time. The effect of reaction and diffusion is observed by studying three test problems. A comparison is performed between the obtained numerical results and some earlier results using L_∞ and relative error norms.     

B-splines with homogenized knots

The use of homogenized knots for manipulating univariate polynomials by blossoming algorithms is extended to piecewise polynomials. A generalization of the B-spline to homogenized knots is studied. The new B-spline retains the triangular blossoming algorithms for evaluation, differentiation and knot insertion. Moreover, the B-spline is locally supported and a Marsden’s identity exists. Spaces of natural splines and certain polynomial spline spaces with more general continuity properties than ordinary splines have bases of B-splines over homogenized knots. Applications to nonpolynomial splines such as trigonometric and hyperbolic splines are made.     

B-splines with homogenized knots

The use of homogenized knots for manipulating univariate polynomials by blossoming algorithms is extended to piecewise polynomials. A generalization of the B-spline to homogenized knots is studied. The new B-spline retains the triangular blossoming algorithms for evaluation, differentiation and knot insertion. Moreover, the B-spline is locally supported and a Marsden’s identity exists. Spaces of natural splines and certain polynomial spline spaces with more general continuity properties than ordinary splines have bases of B-splines over homogenized knots. Applications to nonpolynomial splines such as trigonometric and hyperbolic splines are made.     

Knot insertion algorithms for piecewise polynomial spaces determined by connection matrices

We show that many fundamental algorithms and techniques for B-spline curves extend to geometrically continuous splines. The algorithms, which are all related to knot insertion, include recursive evaluation, differentiation, and change of basis. While the algorithms for geometrically continuous splines are not as computationally simple as those for B-spline curves, they share the same general structure. The techniques we investigate include knot insertion, dual functionals, and polar forms; these prove to be useful theoretical tools for studying geometrically continuous splines.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Approximation by fuzzy B-spline series

We study properties concerning approximation of fuzzy-number-valued functions by fuzzy B-spline series. Error bounds in approximation by fuzzy B-spine series are obtained in terms of the modulus of continuity. Particularly simple error bounds are obtained for fuzzy splines of Schoenberg type. We compare fuzzy B-spline series with existing fuzzy concepts of splines.     

Chebyshev splines beyond total positivity

For polynomial splines as well as for Chebyshev splines, it is known that total positivity of the connection matrices is sufficient to obtain B-spline bases. In this paper we give a necessary and sufficient condition for the existence of B-bases (or, equivalently, of blossoms) for splines with connection matrices and with sections in different four-dimensional extended Chebyshev spaces.     

Quantum B-splines

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q (quantum)-blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden’s identity.     

Shape preserving approximation using least squares splines

Least squares polynomial splines are an effective tool for data fitting, but they may fail to preserve essential properties of the underlying function, such as monotonicity or convexity. The shape restrictions are translated into linear inequality conditions on spline coefficients. The basis functions are selected in such a way that these conditions take a simple form, and the problem becomes non-negative least squares problem, for which effecitive and robust methods of solution exist. Multidimensional monotone approximation is achieved by using tensor-product splines with the appropriate restrictions. Additional inter polation conditions can also be introduced. The conversion formulas to traditional B-spline representation are provided.     

A hybrid method using wavelets for the numerical solution of boundary value problems on the interval

In this work, various aspects of wavelet-based methods for second order boundary value problems under Galerkin framework are investigated. Based on the B-spline multiresolution analysis (MRA) on the line we propose a hybrid method on the interval which combines different treatments for interior and boundary splines. By using this procedure, the MRA structure was conserved and hierarchical representations of the solution at different scales were obtained without much computational effort. Numerical examples are given to verify the effectiveness of the proposed method and the comparison with other techniques is presented.     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

Shape-Constrained Estimation Using Nonnegative Splines

We consider the problem of nonparametric estimation of unknown smooth functions in the presence of restrictions on the shape of the estimator and on its support using polynomial splines. We provide a general computational framework that treats these estimation problems in a unified manner, without the limitations of the existing methods. Applications of our approach include computing optimal spline estimators for regression, density estimation, and arrival rate estimation problems in the presence of various shape constraints. Our approach can also handle multiple simultaneous shape constraints. The approach is based on a characterization of nonnegative polynomials that leads to semidefinite programming (SDP) and second-order cone programming (SOCP) formulations of the problems. These formulations extend and generalize a number of previous approaches in the literature, including those with piecewise linear and B-spline estimators. We also consider a simpler approach in which nonnegative splines are approximated by splines whose pieces are polynomials with nonnegative coefficients in a nonnegative basis. A condition is presented to test whether a given nonnegative basis gives rise to a spline cone that is dense in the space of nonnegative continuous functions. The optimization models formulated in the article are solvable with minimal running time using off-the-shelf software. We provide numerical illustrations for density estimation and regression problems. These examples show that the proposed approach requires minimal computational time, and that the estimators obtained using our approach often match and frequently outperform kernel methods and spline smoothing without shape constraints. Supplementary materials for this article are provided online.     

B-spline solution of a singularly perturbed boundary value problem arising in biology

We use B-spline functions to develop a numerical method for solving a singularly perturbed boundary value problem associated with biology science. We use B-spline collocation method, which leads to a tridiagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical result is found in good agreement with exact solution.     

Shape preserving approximation using least squares splines

Least squares polynomial splines are an effective tool for data fitting, but they may fail to preserve essential properties of the underlying function, such as monotonicity or convexity. The shape restrictions are translated into linear inequality conditions on spline coefficients. The basis functions are selected in such a way that these conditions take a simple form, and the problem becomes non-negative least squares problem, for which effecitive and robust methods of solution exist. Multidimensional monotone approximation is achieved by using tensor-product splines with the appropriate restrictions. Additional inter polation conditions can also be introduced. The conversion formulas to traditional B-spline representation are provided.     

A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection–diffusion problem

A numerical method is proposed for solving singularly perturbed one-dimensional parabolic convection–diffusion problems. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and B-spline collocation method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O((Δx)²+Δt). An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Several numerical experiments have been carried out in support of the theoretical results. Comparisons of the numerical solutions are performed with an upwind finite difference scheme on a piecewise uniform mesh and exponentially fitted method on a uniform mesh to demonstrate the efficiency of the method.     

Tight linear envelopes for splines

Summary. A sharp bound on the distance between a spline and its B-spline control polygon is derived. The bound yields a piecewise linear envelope enclosing spline and polygon. This envelope is particularly simple for uniform splines and splines in Bernstein-Bézier form and shrinks by a factor of 4 for each uniform subdivision step. The envelope can be easily and efficiently implemented due to its explicit and constructive nature. Received February 12, 1999 / Revised version received October 15, 1999 / Published online May 4, 2001     

A B-spline approach for empirical mode decompositions

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem Mathematics subject classification (2000) 94A12. Supported in part by National Aeronautics and Space Administration under grant NAG5-5364, and National Science Foundation under grants NSF0314742 and NSF0312113. Yuesheng Xu: Corresponding author. Supported in part by Natural Science Foundation of China under grant 10371122.     

Design with L-splines

We recently obtained a criterion to decide whether a given space of parametrically continuous piecewise Chebyshevian splines (i.e., splines with pieces taken from different Extended Chebyshev spaces) could be used for geometric design. One important field of application is the class of L-splines, that is, splines with pieces taken from the null space of some fixed real linear differential operator, generally investigated under the strong requirement that the null space should be an Extended Chebyshev space on the support of each possible B-spline. In the present work, we want to show the practical interest of the criterion in question for designing with L-splines. With this in view, we apply it to a specific class of linear differential operators with real constant coefficients and odd/even characteristic polynomials. We will thus establish necessary and sufficient conditions for the associated splines to be suitable for design. Because our criterion was achieved via a blossoming approach, shape preservation will be inherent in the obtained conditions. One specific advantage of the class of operators we consider is that hyperbolic and trigonometric functions can be mixed within the null space on which the splines are based. We show that this produces interesting shape effects.     

Two Families of Approximation Schemes for Delay Systems

Based on the Trotter-Kato approximation theorem for strongly continuous semigroups we develop a general framework for the approximation of delay systems. Using this general framework we construct two families of concrete approximation schemes. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). For the two families we also prove convergence of the adjoint semigroups and uniform exponential stability, properties which are essential for approximation of linear quadratic control problems involving delay systems. The characteristic matrix of the delay system is in both cases approximated by matrices of the same structure but with the exponential function replaced by approximations where Padé fractions in the main diagonal resp. in the diagonal below the main diagonal of the Padé table for the exponential function play an essential role.     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.