Przemysław Gospodarczyk

Przemysław Gospodarczyk, Paweł Woźny

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 order differential equations. Illustrative examples confirm the versatility of our method.

Przemysław Gospodarczyk, Paweł Woźny

In the paper, we give methods of construction of dual bases for the B-spline basis and truncated power basis. Explicit formulas for the dual B-spline basis are obtained using the Legendre-like orthogonal basis of the polynomial spline space presented in (Wei et al., Comput.-Aided Des. 45 (2013), 85-92) and a connection between orthogonal and dual bases of any space given in (Lewanowicz and Woźny, J. Approx. Theory 138 (2006), 129-150). Construction of the dual truncated power basis is performed in two phases. We start with explicit formulas for the dual power basis of the space of polynomials. Then, we expand this basis using an iterative algorithm proposed in (Woźny, J. Comput. Appl. Math. 260 (2014), 301-311). As a result, we obtain the dual truncated power basis. We also present some applications of the proposed dual polynomial spline bases and illustrative examples.

Przemysław Gospodarczyk, Paweł Woźny

In the paper, we show that the transformations between modified Jacobi and Bernstein bases of the constrained space of polynomials of degree at most $n$ can be performed with the complexity $O(n^2)$. As a result, the algorithm of degree reduction of Bézier curves that was first presented in (Bhrawy et al., J. Comput. Appl. Math. 302 (2016), 369--384), and then corrected in (Lu and Xiang, J. Comput. Appl. Math. 315 (2017), 65--69), can be significantly improved, since the necessary transformations are done in those papers with the complexity $O(n^3)$. The comparison of running times shows that our transformations are also faster in practice.

Paweł Woźny, Przemysław Gospodarczyk, Stanisław Lewanowicz

This paper deals with the merging problem of segments of a composite Bézier curve, with the endpoints continuity constraints. We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (P. Woźny, S. Lewanowicz, Comput. Aided Geom. Design 26 (2009), 566--579) to compute the control points of the merged curve. Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging methods.