Monotone cubic interpolation

About: Monotone cubic interpolation is a research topic. Over the lifetime, 1740 publications have been published within this topic receiving 38111 citations.

Quadratic and cubic spline collocation for Volterra integral equations

Abstract: One of the most practical methods for solving Volterra integral equations of the second kind is the polynomial spline collocation method with step-by-step implementation. In the case of quadratic splines this method is stable only for the value c = 1 of the collocation parameter characterizing the position of collocation points between spline knots. It is also known that the cubic and higher order splines give unstable process for any choice of collocation parameter. We replaced an initial condition by a not-a-knot boundary condition at the other end of the interval. Our method is not any more step-by-step method, but it leads to system of equations which can be successfully done by Gaussian elimination. The main advantage which we get is the stability in the whole interval of collocation parameter for quadratic splines and in the interval [1/2, 1] for cubic ones. Main results about stability and convergence are based on the uniform boundedness of corresponding spline interpolation projections. This is due to general convergence theorems for operator equations. In the case c = 1 for quadratic splines and c = 1/2 for cubic ones the norms of collocation projections are of order O(N) (N being the number of used knots) and this allows to get convergence for smooth solutions. We have proved the regular convergence of operators in the case of quadratic splines which implied two-sided error estimates. For quadratic splines it is also shown the convergence in the space of continuously differentiable functions. As well as in the space of continuous functions we get the uniform boundedness of collocation projections for all c ∈ (0, 1) and the projection norms with linear growth for c = 1. The numerical examples support these announced results.

Numerical Simulation of Plane Crack Using Hermite Cubic Spline Wavelet

Abstract: Two-dimensional wavelet-based numerical approximation using Hermite cubic spline wavelet on the interval (HCSWI) is proposed to solve stress intensity factors (SIFs) of plate structures. The good localization property of wavelets is used to approximate displacement fields by multi-scale bases of HCSWI. Example computations are performed for plates with a central crack and double edge cracks. The numerical results prove that, compared with the conventional finite element method and the analytical solutions, the new procedure are efficient in both its accuracy and its reduction of degree of freedoms (DOFs).

Spline solution of non-linear singular boundary value problems

Abstract: A class of non-linear singular boundary value problems is solved by new methods based on non-polynomial cubic spline. We use the quasilinearization technique to reduce the given non-linear problem to a sequence of linear problems. We modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using a non-polynomial cubic spline approximation. Convergence of the methods is shown through standard convergence analysis. Numerical examples are given to illustrate the applicability and efficiency of our methods.

Z-splines: moment conserving cardinal spline interpolation of compact support for arbitrarily spaced data

Abstract: The Z-splines are moment conserving cardinal splines of compact support. They are constructed using Hermite-Birkhoff curves that reproduce explicit finite difference operators computed by Taylor series expansions. These curves are unique. The Z-splines are explicit piecewise polynomial interpolation kernels of cumulative regularity and accuracy. They are succesive spline approximations to the perfect reconstruction filter sinc(x). It is found that their interpolation properties: quality, regularity, approximation order and discrete moment conservation, are related to a single basic concept: the exact representation of polynomials by a long enough Taylor series expansion.

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

Abstract: Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent, however, for the data sets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large data sets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.