On numerical approximation of the Riesz–Caputo operator with the fixed/short memory length

Abstract

In this paper, the Riesz–Caputo operator is studied. This type of fractional operator is a combination of the left and right Caputo derivatives. The series representation of the analyzed fractional operators with fixed memory length is presented. In the main part of the paper, three modified methods of numerical integration are applied for the approximation of the left and right Caputo, and Riesz–Caputo derivatives. Numerical schemes based on three types of interpolating functions (constant, linear and quadratic function) are presented. The in-depth numerical analysis of the presented schemes is conducted. Absolute errors and experimental rates of convergence, for the considered methods, are calculated and presented.

Keywords

• Fractional derivatives
• Numerical schemes
• Caputo operator
• Riesz–Caputo operator
• Fixed memory length