Algorithms and Software for Calculating the Spectral Characteristic of the Fractional-order Integrodifferential Operator by Haar Functions

Abstract

The article objective is to describe the algorithms and software for calculating the spectral characteristic of the fractional-order integro-differentiation operator by the orthonormal basis of Haar functions. These results are necessary to have a spectral form of mathematical description for control systems. They can be used to solve analysis, synthesis and identification problems using the spectral method for deterministic and stochastic control systems, whose mathematical models are described by equations with fractional derivatives. The obtained results extend the previous authors’ papers, which consider the algorithms and software for calculating the spectral characteristics of various linear operators. These operators are multiplication, differentiation and integration operators, fractional-order differentiation and integration operators, time-inverse and time-lag operators, etc. The following orthonormal and bi-orthonormal functions were used: Legendre and Chebyshev polynomials, Laguerre and Hermite polynomials, trigonometric functions, Walsh and Haar functions, Faber–Schauder functions, function based on wavelets, functions generated by splines.The article represents, as the final result, not only the algorithms and software, but also a technique to derive desirable relationships in Mathcad for calculating the spectral characteristics. This technique can be applied to other basis systems as well. Such algorithms and software extend the specialized expansion package for Mathcad and can be adapted for Maple, Mathematica, and Matlab, as well as for the Spectrum software.To test the developed algorithms and software for calculating the spectral characteristic of the fractional-order integro-differentiation operator by the orthonormal basis of Haar functions we consider the fractional differentiation and integration of the unit step function and the Abel integral equation.