An approximation of the fractional integrals using quadratic interpolation

Abstract

In this paper we present a numerical scheme to calculations of the left fractional integral. To calculate it we use the fractional Simpson's rule (FSR). The FSR is derived by applying quadratic interpolation. We calculate errors generated by the method for particular functions and compare the obtained results with the fractional trapezoidal rule (FTR).

Numerical approximation of the Riemann-Liouville fractional integrals using the Akima spline interpolation

“…Analysing the literature (e.g. [10][11][12][13][14][15][16]), it can be noticed that better (with high accuracy, fast convergence) approximation methods for the fractional integrals and derivatives are constantly being developed. Typically, two main steps need to be used to develop a numerical method of integration.…”

Numerical Approximations of the Riemann–Liouville and Riesz Fractional Integrals

“…Among the pioneering works in this field, the book by Oldham and Spanier (1974) can be distinguished. The reviews of different numerical methods (so far known) for fractional integrals and derivatives can be found in Cai and Li (2020), Li and Zeng (2015), Almeida et al (2015), Blaszczyk and Siedlecki (2014), Baleanu et al (2012), Malinowska et al (2015), Blaszczyk et al (2018), Odibat (2006), Dimitrov (2021), Budak et al (2023). Due to the wide applications of fractional calculus, it is becoming increasingly important to develop numerical algorithms with high accuracy, fast convergence and less storage memory.…”

Numerical algorithms for approximation of fractional integral operators based on quadratic interpolation

“…The ability to calculate these types of fractional operators is very important to get a graphical interpretation of solutions of fractional differential equations of the variational type, because the analytical evaluations for any function are difficult to achieve. Bearing in mind the abovementioned facts, many authors propose different approaches to discretization and numerical evaluation of the fractional or integer order operators …”