Related papers: Splines and Fractional Differential Operators
Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose…
In this article, we consider some generalizations of polynomial and exponential B-splines. Firstly, the extension from integral to complex orders is reviewed and presented. The second generalization involves the construction of uncountable…
We extend the concept of exponential B-spline to complex orders. This extension contains as special cases the class of exponential splines and also the class of polynomial B-splines of complex order. We derive a time domain representation…
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…
Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar…
Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree…
We analyse sampling and average sampling techniques for fractional spline subspaces of $L^{2}({\mathbb{R}}).$ Fractional B-splines $\beta_{\sigma}$ are extensions of Schoenberg's polynomial splines of integral order to real order $\sigma >…
Nowadays, fractional differential equations are a well established tool to model phenomena from the real world. Since the analytical solution is rarely available, there is a great effort in constructing efficient numerical methods for their…
We introduce discrete analogues of the exponential, sine, and cosine functions. Then using a discrete trigonometric version of a non-polynomial divided difference, we define discrete analogues of the trigonometric B-splines. We derive a…
In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the…
Pseudo-splines of integer order $(m,\ell)$ were introduced by Daubechies, Han, Ron, and Shen as a family which allows interpolation between the classical B-splines and the Daubechies' scaling functions. The purpose of this paper is to…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
This paper reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by (a…
The novel concept of box spline of complex degree is introduced and several of its properties derived and discussed. These box splines of complex degree generalize and extend the classical box splines. Relations to a class of fractional…
In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by…
Spline functions have long been used in numerically solving differential equations. Recently it revives as isogeometric analysis, which uses NURBS for both parametrization and element functions. In this paper, we introduce some multivariate…
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…
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…