Une étude des splines non polynomiales et leurs applications numériques

Abstract

Subdivision schemes are recursive methods for generating smooth functions from discrete data. These schemes have received considerable attention in the last decades because they allow designs of efficient, local, and hierarchical modeling algorithms in a wide range of applications related to computer-aided geometric design, computer graphics, and the recent area of interest like fractional calculus. A subdivision starts with a set of initial control points and then recursively generates denser control points by a linear combination of nodes of a lower refinement level. Spline functions and subdivision schemes are strongly linked. To be more specific, several authors widely discussed smooth subdivision schemes based on polynomial splines in the literature. Accordingly, we have devoted this Ph.D. thesis to providing an in-depth theoretical and practical study of subdivision schemes based on non-polynomial splines, i.e., cycloidal splines. Namely, we have considered the splines spanned by polynomials and trigonometric or hyperbolic functions. This Ph.D. thesis is divided into two parts: The first section is about hyperbolic algebraic splines. Indeed, uniform algebraic hyperbolic B-splines of any degree 𝑘 have been examined. Then, for each arbitrary order 𝑘, we established a general formula for the refinement equations. The subdivision schemes are established using these refinement equations. Finally, we provided a new multiresolution technique for curves of general topology to introduce a new inverse subdivision scheme connected with algebraic hyperbolic B-splines of order 𝑘 = 3. Numerical results back up theoretical conclusions. We proposed a new algebraic trigonometric Hermite interpolation operator in the second part. This operator interpolates the function values and the first derivative values at the break-points of a partition. The considered Hermite interpolating splines give an optimal convergence order and produce linear polynomials and trigonometric functions. Hence, quadrature rules with endpoint corrections are provided based on integrating the considered Hermite interpolating splines. These rules are employed to solve the 1-D Fredholm integral equations numerically. The error bound is examined, and numerical examples of the proposed interpolating splines' performance are provided.