# Approximation of the Riesz-Caputo Derivative by Cubic Splines
## 1 Introduction  
In the last few decades, differential equations with fractional derivatives have emerged as a main tool for constructing models of real-world phenomena closer to experimental observations. Nowadays, models using fractional derivatives in time and/or in space are commonly used in various fields of sciences, such as biology, physics, mechanics, economics, control theory, just to cite a few (see, for instance, [1; 2; 3; 4; 5; 6; 7; 8; 9] and references therein). The reason lies in the fact that the fractional derivative is a nonlocal operator and it allows to approach the memory and hereditary properties of physical systems in a clear way.  
In the field of geophysics, fractional differential equations are widely used for modeling anomalous diffusion in porous media, a phenomenon in which particles have been observed to spread at a rate which is incompatible with classical Brownian motion. In this context, the Riesz fractional derivative has been shown to be more attractive compared to the left-sided derivative since it is able to take into account contributions from both sides of the domain [2; 10]. In fact, the Riesz derivative is a linear combination of the left- and right-sided derivatives, resulting in a fractional derivative centrally symmetric on finite domains. Although in literature the Riesz derivative defined through the left and right Riemann-Liouville derivatives is commonly used to model fractional diffusion, the Riesz-Caputo derivative seems more suitable for avoiding nonphysical issues (see [11]). For this reason, in this paper, we consider boundary value problems with the Riesz-Caputo derivative in space. In particular, the aim of the paper is to construct an approximation of the Riesz-Caputo derivative by cubic splines and use this approximation to solve fractional differential boundary value problems (FBVPs) using the spline collocation method introduced in [12; 13]. The literature offers several numerical methods to approximate the Riesz derivative--finite difference methods [14; 15; 16; 17], finite element methods [18; 19; 20], spectralmethods [21; 22; 23], mesh-free methods [24; 25]--while the use of spline approximations is still limited [26; 27; 28]. The main contribution of this paper is to give the explicit expression of the cubic spline approximation for the Riesz-Caputo derivative of a given function and to show its effectiveness in solving FBVPs using the spline collocation method. The choice of cubic splines, instead of linear or quadratic splines, was made because they yield smooth approximations while maintaining a low computational cost. To make the evaluation of the Riesz-Caputo derivative more efficient, we exploit the symmetry of the spline basis and avoid in this way the computation of the right Caputo derivative of the basis functions.  
To the best of our knowledge, this is the first time that cubic splines have been used in the context of Riesz-Caputo derivatives.  
The paper is organized as follows. In Section 2.1, we describe the fractional differential problem we are interested in. In Section 2.2, we recall some basic properties of the cubic B-spline and of the cubic spline basis on finite intervals. Section 2.3 is devoted to the approximation of the Riesz-Caputo derivative, while the collocation method is described in Section 2.4. To show the accuracy of the numerical approximation, in Section 3, we solve some FBVPs using the spline collocation method. Finally, in Section 4, we discuss conclusions and possible future developments.