Fractional-order system dynamics
Application of fractional calculus concerning complex orders has gained impetus across diverse domains owing to its flexibility and broad peculiarities in systems modeling, analysis, control, stability (exponential stability, h-infinity stability, and so forth). The inherent robustness and compact representation of the systems or controllers as fractional order system yield significant benefits in improving the integer-order control. Concerning fractional differential equations (i.e.: partial differential equations (PDEs), ordinary differential equations (ODEs), difference equations (DEs), and so on), the analyses, computations and machine learning applications of fractional calculus bring about upper hands in different fields of practice by providing adequate instruments that characterize nonlocality and heredity.
Fractional calculus, addressing derivatives and integrals of arbitrary order, develops the theory of differentiation and integration of real or complex order, extending the fundamental operations of classical calculus to fractional orders while examining the solving of differential equations’ methods. Fractional derivatives and fractional integrals have their applications in modeling of anomalous diffusion, different physical phenomena and relaxation processes. Fractional differentiation enables the computation of derivatives of non-integer orders, whereas traditional deals with integer orders, applications found in engineering, physics, and other domains whose systems exhibit non-classical behaviors and patterns. As for the relationship between differentiation and fractional calculus, it can be noted that fractional differentiation is a natural extension of classical differentiation, and it enables a way for describing complex behaviors in various systems so that non-integer orders and applications thereof can be explored profoundly and applicably.
Fractional-order with its applications, as a generalization of integer-order elements, considers the memory effect of the systems where the order derivatives may be distributed, variable and / or complex. Since holding memory effects, the provision of an exact discretion tool for theoretical analyses and applications might be challenging. Thus, theoretical and numerical work on fractional differential equations encompass novel application of fractional-order calculus of complex orders, control systems, fractional dynamics of complex orders along with signal / image processing, system identification, observability, controllability and biomedical applications in applied sciences, engineering, basic sciences, among many others.
Regarding fractional calculus and interdisciplinary applications thereof, the following topics can be of interest:
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