Fractional Order Derivative and Integral Computation with a Small Number of Discrete Input Values Using Grünwald–Letnikov Formula

Abstract

High-accuracy computer approximation of fractional derivatives and integrals by applying Grünwald–Letnikov formula generally requires a large number of input values. If required amount cannot be supplied, accuracy of approximation drops drastically. In this paper, we solve a difficult problem in this scope, i.e., when input data consists only of a small number of discrete values. Furthermore, some of the values may be unusable for computational purposes. Our problem solution includes an appropriate method of input data preprocessing, an interpolation algorithm with extrapolation abilities, a central point function discretization schema, recurrent computational method of coefficients and application of Horner’s schema for the core of the Grünwald–Letnikov method: coefficients and function’s values multiplication. Numerical method presented in the paper enables to compute fractional derivatives and integrals of complicated functions with much higher accuracy than it is possible when application of the default approach to Grünwald–Letnikov method computer implementation is applied. This new method usually takes only 10% of function’s values required by the default approach for the same computations and is much less restrictive for their quality. The general novelty of the method is an efficient configuration of existing numerical methods and enhancement of their abilities by applying modern programming language — Python and arbitrary precision arithmetic for computations.