Quintic Spline Solutions of Fourth Order Boundary-Value Problems

A general formula is presented for any order derivative of Chebyshev polynomials instead of the existing recursive relationship. Hence, the Chebyshev finite difference method is made applicable not only to second order problems but also to higher order boundary value problems. The generalized method is applied to a variety of higher order boundary value problems and it is seen that the obtained results are more accurate than the other numerical methods in absolute error.

The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. Determination of the optimum value of the extension parameter can be achieved by scanning some intervals containing zero. This study aims to solve some initial boundary value problems con- structed for the Gardner equation by the extended cubic B-spline collocation method.The test problems are derived from some analytical studies t...

This paper presents the non-linear generalization of a previous work on matrix differential models. It focusses on the construction of approximate solutions of first-order matrix differential equations Y'(x)=f(x,Y(x)) using matrix-cubic splines. An estimation of the approximation error, an algorithm for its implementation and illustrative examples for Sylvester and Riccati matrix differential equations are given.

Practically, for all real measuring devices the result of a measurement is a convolution of an input signal with a hardware function of a unit {\phi}. We call a spline to be {\phi}-interpolating if the convolution of an input signal with a hardware function of a unit {\phi} coincides with the convolution of the spline with the hardware function. In the following article we consider conditions imposed on the hardware function {\phi} under which a second- and third-order {\phi}...

Splines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, B\'ezier and Catmull-Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic B\'ezier and Catmull-Rom curve segments, rather than going through their properties. By d...

Some physical applications of the Passare-Tsikh solution of a principal quintic equation are discussed. As an example, a quintic equation of state is solved in detail. This approach provides analytical approximations for several problems admitting until now only numerical solutions.

The paper is devoted to problem of spline approximation. A new method of nodes location for curves and surfaces computer construction by means of B-splines and results of simulink-modeling is presented. The advantages of this paper is that we comprise the basic spline with classical polynomials both on accuracy, as well as degree of paralleling calculations are also shown.

This paper investigates some univariate and bivariate constrained interpolation problems using rational quartic fractal interpolation functions, which has been submitted long back in a reputed journal and revised as per the journal requirement. This research is extension of the work [S. K. Katiyar and A. K. B. Chand, Shape Preserving Rational Quartic Fractal Functions, Fractal, in Press].

In this paper I uncover and explain---using contour integrals and residues---a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Pad\'e scheme for generating fourth-order accurate compact finite differences gives \textsl{exactly} the derivatives at the interior nodes needed to guarantee twice-continuous differentiability for cubic splines. %I found this connection surprising, because the two...

In this study collocation method based on the extended B-spline functions for the numerical solutions of the Generalized Burhers Fisher equation is set up. The approximate solution of the equation is constructed with the combination of the extended B-splines. Some initial boundary value problems are solved by the proposed method. The accuracy and validity and of the method is demonstrated by measuring the error between the numerical solutions and the analytical solutions, if ...