Construction of non-polynomial splines of the first level with fourth order of approximation
25 Nov 2020-Vol. 2293, Iss: 1, pp 420016
TL;DR: In this article, the authors proposed an approximation with the Hermite polynomial cubic splines with the fourth order of approximation, which is called first level approximation and has the properties of polynomials and trigonometric functions.
Abstract: Interpolation using Hermite polynomial cubic splines is well known and often used. Here we propose an approximation with the non-polynomial splines with the fourth order of approximation. The splines uses the values of the function and the first derivative of the function in the nodes. We call the approximation as first level approximation because it uses the first derivative of the function. This approximation has the properties of polynomial and trigonometric functions. Here we also have constructed a non-polynomial interpolating spline which has continuous the first and second derivative. This approximation uses the values of the function at the nodes and the values of the first derivative of the function at the ends of the interval [a, b]. Estimates of the approximations are given and the constants included in them are calculated. Numerical examples are given.
References
More filters
TL;DR: In this article, a direct numerical approach for fractional advection-diffusion equations (ADEs) is proposed using a set of cubic trigonometric B-splines as test functions.
Abstract: This article studies a direct numerical approach for fractional advection–diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a differential quadrature (DQ) method is firstly proposed for the 1D and 2D time-fractional ADEs of order (0, 1]. The weighted coefficients are determined, and with them, the original equation is transformed into a group of general ordinary differential equations (ODEs), which are discretized by an effective difference scheme or Runge–Kutta method. The stability is investigated under a mild theoretical condition. Secondly, based on a set of cubic B-splines, we develop a new Crank–Nicolson type DQ method for the 2D space-fractional ADEs without advection. The DQ approximations to fractional derivatives are introduced, and the values of the fractional derivatives of B-splines are computed by deriving explicit formulas. The presented DQ methods are evaluated on five benchmark problems and the simulations of the unsteady propagation of solitons and Gaussian pulse. In comparison with the algorithms in the open literature, numerical results finally illustrate the validity and accuracy.
19 citations
TL;DR: In this article, an application of the quartic trigonometric B-spline finite element method is used to solve the regularized long wave equation numerically, and the accuracy of the proposed methods are demonstrated by test problems and numerical results are compared with the exact solution.
Abstract: In this paper, an application of the quartic trigonometric B-spline finite element method is used to solve the regularized long wave equation numerically. This approach involves a Galerkin method based on the quartic trigonometric B-spline function in space discretization together with second and fourth order schemes in time discretization. The accuracy of the proposed methods are demonstrated by test problems and numerical results are compared with the exact solution and some previous results.
16 citations
26 May 2009
TL;DR: This paper deals with computation and utilisation of auxiliary points in order to further increase the ability of RBF to restore damaged areas in image and achieves the best possible results in acceptable time of computation.
Abstract: Utilisation of Radial Basis Functions (RBF) for reconstruction of damaged images became common technique nowadays. This paper deals with computation and utilisation of auxiliary points in order to further increase the ability of RBF to restore damaged areas in image. Our goal was to achieve the best possible results in acceptable time of computation. We put stress mainly on cases, where the image is heavily damaged e.g. by extreme noise. In these cases our new proposed approach achieved very usable results that even surpassed our expectations.
16 citations
TL;DR: In this paper, a non-polynomial B-spline interpolant and a weighted scheme for space and time discretization is proposed for solving PHI-Four and Allen-Cahn equations.
Abstract: In this paper, we develop a numerical solution based on nonpolynomial B-spline (trigonometric B-spline) collocation method for solving time-dependent equations involving PHI-Four and Allen–Cahn equations. A three-time-level implicit algorithm has been derived. This algorithm combines the trigonometric B-spline interpolant and the \(\theta \)-weighted scheme for space and time discretization, respectively. Convergence analysis is discussed and the accuracy of the presented method is \(O( {\tau ^{2}+h^{2}} ).\) Applying von Neumann stability analysis, the proposed technique is shown to be unconditionally stable. Three test problems are demonstrated to reveal that our method is reliable, efficient and very encouraging.
14 citations
17 Oct 2013
TL;DR: A novel and fast, simple and robust algorithm with O(N) expected complexity which enables to decrease run-time needed to find an exact maximum distance of two points in E2.
Abstract: This paper describes novel and fast, simple and robust algorithm with O(N) expected complexity which enables to decrease run-time needed to find an exact maximum distance of two points in E2. The proposed algorithm has been evaluated experimentally on larger different datasets. The proposed algorithm gives a significant speed-up to applications, when medium and large data sets are processed. It is over 10 000 times faster than the standard algorithm for 106 points randomly distributed points in E2. Experiments proved the advantages of the proposed algorithm over the standard algorithm and convex hull diameters approaches.
7 citations