Analysis, Modeling and Stability of Fractional Order Differential Systems 1

3Comparison of Two Simulation Techniques

3.1. Introduction

Solving FDEs has been a fundamental objective of fractional calculus for a long time. Analytical solutions are available only for a limited number of theoretical cases [OLD 72, SAM 93, MIL 93, POD 99]. The need for practical solutions for fractional calculus applications in various domains of engineering has stimulated the development of numerical algorithms. Many techniques are available for the simulation of fractional order equations and systems, such as methods based on Laplace and Fourier transforms [POD 99], SVD decomposition [TRI 10a], direct solutions based on the Grünwald–Letnikov approximation [POD 99, ORT 11], the matrix approximation method [POD 00], the diffusive representation [HEL 00], approximate state space representations [POI 03] and various numerical methods (see, for example, [MON 10, PET 11]). The accuracy of numerical algorithms was particularly addressed by Diethelm [DIE 02, DIE 08, DIE 10]. In [AGR 07], the authors propose a comparison of different techniques, particularly Diethelm, Grünwald and fractional integrator. Diethelm techniques are more accurate, but they require large computation time; the Grünwald approach performs a good compromise between precision and computation time, whereas the integrator approach is faster, with medium precision.

Among the different techniques proposed in the technical literature, the more popular one and at the same time the more simple one is certainly ...