Abstract
In this paper, the stability of fractional differential equations (FDEs) with unknown parameters is studied. Using the graphical based D-decomposition method, the parametric stability analysis of FDEs is investigated without complicated mathematical analysis. To achieve this, stability boundaries are obtained firstly by a conformal mapping from s-plane to parameter space composed by unknown parameters of FDEs, and then the stability region set depending on the unknown parameters is found. The applicability of the presented method is shown considering some benchmark equations, which are often used to verify the results of a new method. Simulation examples show that the method is simple and give reliable stability results.
Highlights
Fractional differential equations [18, 28] are generalization of classical integer-order differential equations through the application of fractional calculus [30], which has been developed by pure mathematicians firstly since after half of the 19th century, though engineers and physicists found applications of fractional calculus for various concepts 100 years later [50]
Motivated by the need of stability analysis for fractional differential equations (FDEs) with unknown parameters, we suggest in this paper an efficient graphical based stability analysis using the D-decomposition method [21]
A graphical based stability analysis method is presented for FDEs
Summary
Fractional differential equations [18, 28] are generalization of classical integer-order differential equations through the application of fractional calculus [30], which has been developed by pure mathematicians firstly since after half of the 19th century, though engineers and physicists found applications of fractional calculus for various concepts 100 years later [50]. These methods which are implemented with the advancement of high speed computers aimed to better and better characterization, design tools and control c Vilnius University, 2019 performance of modern technological products of engineering systems of developing civilization These developments, which had covered only static system models involving geometry and algebra until 1965, had started using dynamical models involving differential and integral calculus since 1965; and have been accelerating since the 1960’s with the fractional-order modelling involving FDEs, which have gained force and dare with high speed computers. FDEs have become a powerful tool in studying, designing and control of physical systems and engineering products of the present world and it still constitutes a popular research area resulting with new definitions of fractional derivative and its applications in need.
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