Algorithms and software for calculating the spectral characteristic of the fractional-order integro-differential operator by Walsh functions

Abstract

The paper describes algorithms and software for calculating the spectral characteristic of the fractional-order integro-differential operator by the orthonormal basis of Walsh functions applied in the spectral form of mathematical description of operating systems. Algorithms and programs for calculating the spectral characteristics of the fractional-order integro-differential operator are formed by changing the basis using the corresponding algorithms and programs for calculating the spectral characteristic of the fractional-order integro-differential operator by the orthonormal basis of Haar functions. The paper contains not only the final result as algorithms and programs, but also gives a technique of deriving the necessary relationships in Mathcad for calculating the spectral characteristics, which can be applied to other orthonormal basis. Algorithms and software for calculating the spectral characteristics of the integro-differential operator can be used to solve different problems such as analysis, synthesis and identification of control systems whose mathematical models are described by equations with fractional-order derivatives. The paper provides the necessary information about the spectral form of the mathematical description for control systems, the definition of the fractional-order integro-differential operator, as well as Haar and Walsh functions. The developed programs for Mathcad are given as source listings. The problem of solving the Abel integral equation is considered as an example. The calculation results obtained by the spectral method with algorithms and software for calculating the spectral characteristic of the fractional-order integro-differential operator by the orthonormal basis of Walsh functions correspond to the exact solution.