Quintic Spline Solutions of Fourth Order Boundary-Value Problems

In the context of cubic splines, the authors have contributed to a recent paper dealing with the computation of nonlinear derivatives at the interior nodes so that monotonicity is enforced while keeping the order of approximation of the spline as high as possible. During the review process of that paper, one of the reviewers raised the question of whether a cubic spline interpolating monotone data could be forced to preserve monotonicity by imposing suitable values of the fir...

In this paper we present a unified method for solving general polynomial equations of degree less than five.

In this work we present a method, based on the use of Bernstein polynomials, for the numerical resolution of some boundary values problems. The computations have not need of particular approximations of derivatives, such as finite differences, or particular techniques, such as finite elements. Also, the method doesn't require the use of matrices, as in resolution of linear algebraic systems, nor the use of like-Newton algorithms, as in resolution of non linear sets of equatio...

In the classical theory of cubic interpolation splines there exists an algorithm which works with only $O\left( n\right)$ arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces their computation to interpolation cubic splines and also performs with $O\left( n\right)$ arithmetic operations. In this paper it is shown that many features of the polynomial cubic spline setting carry over to the larger class of $L$-spline...

One of the main purposes of this article is to give functional equations and differential equations between Bernstein basis functions and generating functions of B-spline curves. Using these equations, very useful formulas containing the relationships among the uniform B-spline curves, the Bernstein basis functions, and other special numbers and polynomials are derived. By applying p-adic integrals to these polynomials, many novel formulas are also derived. Furthermore, by ap...

Computing accurate splines of degree greater than three is still a challenging task in today's applications. In this type of interpolation, high-order derivatives are needed on the given mesh. As these derivatives are rarely known and are often not easy to approximate accurately, high-degree splines are difficult to obtain using standard approaches. In Beaudoin (1998), Beaudoin and Beauchemin (2003), and Pepin et al. (2019), a new method to compute spline approximations of ...

In this article, we study the numerical solution of the one dimensional nonlinear sine-Gordon by using the modified cubic B-spline differential quadrature method. The scheme is a combination of a modified cubic B spline basis function and the differential quadrature method. The modified cubic B spline is used as a basis function in the differential quadrature method to compute the weighting coefficients. Thus, the sine Gordon equation is converted into a system of ordinary di...

A systematic construction of higher order splines using two hierarchies of polynomials is presented. Explicit instructions on how to implement one of these hierarchies are given. The results are limited to interpolations on regular, rectangular grids, but an approach to other types of grids is also discussed.

Neville's algorithm is known to provide an efficient and numerically stable solution for polynomial interpolations. In this paper, an extension of this algorithm is presented which includes the derivatives of the interpolating polynomial.

In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties of dual Bernstein polynomials which guarantee high efficiency of our approach. The method can deal with both linear and nonlinear differential equations. Moreover, not only second order differential equations can be solved but also higher or...