Abstract:
We provide numerical solution to the one-dimensional wave equations in polar coordinates, based on the cubic B-spline quasi-interpolation. The numerical scheme is obtained by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a forward difference to approximate the time derivative of the dependent variable. The accuracy of the proposed method is demonstrated by three test problems. The results of numerical experiments are compared with analytical solutions by calculating errors -norm and -norm. The numerical results are found to be in good agreement with the exact solutions. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. 1. Introduction The term “spline” in the spline function arises from the prefabricated wood or plastic curve board, which is called spline, and is used by the draftsman to plot smooth curves through connecting the known point. The use of spline function and its approximation play an important role in the formation of stable numerical methods. As the piecewise polynomial, spline, especially B-spline, have become a fundamental tool for numerical methods to get the solution of the differential equations. In the past, several numerical schemes for the solution of boundary value problems and partial differential equations based on the spline function have been developed by many researchers. As early in 1968 Bickley [1] has discussed the second-order accurate spline method for the solution of linear two-point boundary value problems. Raggett and Wilson [2] have used a cubic spline technique of lower order accuracy to solve the wave equation. Chawla et al. [3] solved the one-dimensional transient nonlinear heat conduction problems using the cubic spline collocation method in 1975. Rubin and Khosla [4] first proposed the spline alternating direction implicit method to solve the partial differential equation using the cubic spline and enhanced accuracy of the approximate solution of the second derivative to the same as that of the first derivative. Jain and Aziz [5] have derived fourth-order cubic spline method for solving the nonlinear two-point boundary value problems with significant first derivative terms. In recent years, El-Hawary and Mahmoud [6], Mohanty [7], Mohebbi and Dehghan [8], Zhu and Wang [9], Ma et al. [10], Dosti and Nazemi [11], Wang et al. [12], and other researchers [13–16] have derived various numerical methods for solution of partial differential equations based on the spline function. The hyperbolic

Abstract:
The aim of this study is to describe the formulation of Quarter-Sweep Modified Successive Over-Relaxation (QSMSOR) iterative method using cubic polynomial spline scheme for solving second order two-point linear boundary value problems. To solve the problems, a linear system will be constructed via discretization process by using cubic spline approximation equation. Then the generated linear system has been solved using the proposed QSMSOR iterative method to show the superiority over Full-Sweep Modified Successive Over-Relaxation (FSMSOR) and Half-Sweep Modified Successive Over-Relaxation (HSMSOR) methods. Computational results are provided to illustrate the effectiveness of the proposed method.

Abstract:
From the result in [1] it follows that there is a unique quadratic spline which bounds the same area as that of the function. The matching of the area for the cubic spline does not follow from the corresponding result proved in [2]. We obtain cubic splines which preserve the area of the function.

Abstract:
Based on analysis of basic cubic spline interpolation, the clamped cubic spline interpolation is generalized in this paper. The methods are presented on the condition that the first derivative and second derivative of arbitrary node are given. The Clamped spline and Curvature-adjusted cubic spline are also generalized. The methods are presented on the condition that the first derivatives of arbitrary two nodes or second derivatives of arbitrary two node are given. At last, these calculation methods are illustrated through examples.

Abstract:
We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in -direction and finite difference approximation in -direction for the numerical solution of one-dimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method. 1. Introduction We consider the one-dimensional wave equation in polar forms: with the following initial conditions: and the following boundary conditions: where and . We assume that the conditions (2) and (3) are given with sufficient smoothness to maintain the order of accuracy in the numerical method under consideration. The study of wave equation in polar form is of keen interest in the fields like acoustics, electromagnetic, fluid dynamics, mathematical physics, and so forth. Efforts are being made to develop efficient and high accuracy finite difference methods for such types of PDEs. During the last three decades, there has been much effort to develop stable numerical methods based on spline approximations for the solution of time-dependent partial differential equations. But so far in the literature, very limited spline methods are there for the wave equation in polar coordinates. In 1968-69, Bickley [1] and Fyfe [2] studied boundary value problems using cubic splines. In 1973, Papamichael and Whiteman [3], and the next year, Fleck [4] and Raggett and Wilson [5] have used a cubic spline technique of lower order accuracy to solve one-dimensional heat conduction equation and wave equation, respectively. Then, Jain et al. [6–9] have derived cubic spline solution for the differential equations including fourth order cubic spline method for solving the nonlinear two point boundary value problems with significant first derivative terms. Recently, Kadalbajoo et al. [10, 11] and Khan et al. [12, 13] have studied parametric cubic spline technique for solving two point boundary value problems. In recent years, Rashidinia et al. [14], Ding and Zhang [15], and Mohanty et al. [16–21] have discussed spline and high order finite difference methods for the solution of hyperbolic equations. In this present paper, we follow the idea of Jain and Aziz [7] by using nonpolynomial spline in compression approximation to develop order four method in space direction for the wave equation in polar co-ordinates. We have shown that our method is in general

Abstract:
According to the characteristics of the closed curve on every section of mannequin, the rectangular coordinates function was converted to the polar coordinates function,cubic spline interpolation conducted,and x, y, z value were converted later again.This method is simple,with less calculation.But fitting model of the shoulder and neck deformated greatly.After analyzing of the reason of large deformation,the parametric cubic spline interpolation method was used for x, y, z respectively on Warps and wefts wi...

Abstract:
Based on analysis of cubic spline interpolation, the differentiation formulas of the cubic spline interpolation on the three boundary conditions are put up forward in this paper. At last, this calculation method is illustrated through an example. The numerical results show that the spline numerical differentiations are quite effective for estimating first and higher derivatives of equally and unequally spaced data. The formulas based on cubic spline interpolation solving numerical integral of discrete function are deduced. The degree of integral formula is n=3.The formulas has high accuracy. At last, these calculation methods are illustrated through examples.

Abstract:
In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x_0,x_N], the interpolating Cubic Spline (SFIF) and their derivatives converge respectively to the data generating function and its derivatives at the rate of h^(2-j+e) (0

Abstract:
We define a new basis of cubic splines such that the coordinates of a natural cubic spline are sparse. We use it to analyse and to extend the classical Schoenberg and Reinsch result and to estimate a noisy cubic spline. We also discuss the choice of the smoothing parameter. All our results are illustrated graphically.

Abstract:
In the present paper, we obtain an asymptotically precise estimate for the derivative of the difference between the cubic spline interpolating at the mid points of a uniform partition and the function interpolated.