Book

# Functional Fractional Calculus

01 Jun 2011-

TL;DR: In this article, a modern approach to solve the solvable system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving approximately exact series solutions.

AbstractWhen a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with ordinary differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematical and geometrical explanations, but also several practical applications are given particularly for system identification, description and then efficient controls. The normal physical laws like, transport theory, electrodynamics, equation of motions, elasticity, viscosity, and several others of are based on ordinary calculus. In this book these physical laws are generalized in fractional calculus contexts; taking, heterogeneity effect in transport background, the space having traps or islands, irregular distribution of charges, non-ideal spring with mass connected to a pointless-mass ball, material behaving with viscous as well as elastic properties, system relaxation with and without memory, physics of random delay in computer network; and several others; mapping the reality of nature closely. The concept of fractional and complex order differentiation and integration are elaborated mathematically, physically and geometrically with examples. The practical utility of local fractional differentiation for enhancing the character of singularity at phase transition or characterizing the irregularity measure of response function is deliberated. Practical results of viscoelastic experiments, fractional order controls experiments, design of fractional controller and practical circuit synthesis for fractional order elements are elaborated in this book. The book also maps theory of classical integer order differential equations to fractional calculus contexts, and deals in details with conflicting and demanding initialization issues, required in classical techniques. The book presents a modern approach to solve the solvable system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving approximately exact series solutions.Historically, Sir Isaac Newton and Gottfried Wihelm Leibniz independently discovered calculus in the middle of the 17th century. In recognition to this remarkable discovery, J.von Neumann remarked, the calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more equivocally than anything else the inception of modern mathematical analysis which is logical development, still constitute the greatest technical advance in exact thinking.This XXI century has thus started to think-exactly for advancement in science & technology by growing application of fractional calculus, and this century has started speaking the language which nature understands the best.

##### Citations
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01 Jan 2012

413 citations

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Journal ArticleDOI
TL;DR: This review investigates its progress since the first reported use of control systems, covering the fractional PID proposed by Podlubny in 1994, and is presenting a state-of-the-art fractionalpid controller, incorporating the latest contributions in this field.
Abstract: Fractional calculus has been studied for over three centuries, and it has multifarious applications in science and engineering. This review investigates its progress since the first reported use of control systems, covering the fractional PID proposed by Podlubny in 1994, and is presenting a state-of-the-art fractional PID controller, incorporating the latest contributions in this field. It highlights developments in the field of fractional PID controllers, including their design and tuning, as well as explores their various versions. Software tools associated to the design of fractional PID controllers are also discussed.

309 citations

Journal ArticleDOI
TL;DR: In this paper, a numerical approach to the immunogenetic tumour model using differential and integral operators with Mittag-Leffler law was made, where fractional Atangana- Baleanu derivative has been utilized in the structure of proposed model.
Abstract: Mathematical biology is one of the interesting research area of applied mathematics that describes the accurate description of phenomena in biology and related health issues. The use of new mathematical tools and definitions in this area of research will have a great impact on improving community health by controlling some diseases. This is the best reason for doing new research using the latest tools available to us. In this work, we will make novel numerical approaches to the immunogenetic tumour model to using differential and integral operators with Mittag-Leffler law. To be more precise, the fractional Atangana- Baleanu derivative has been utilized in the structure of proposed model. This paper proceeds by examining and proving the convergence and uniqueness of the solution of these equations. The Adam Bashforth’s Moulton method will then be used to solve proposed fractional immunogenetic tumour model. Numerical simulations for the model are obtained to verify the applicability and computational efficiency of the considered process. Similar models in this field can also be explored similarly to what has been done in this article.

194 citations

Journal ArticleDOI
TL;DR: This paper investigates the operation of a hybrid power system through a novel fuzzy control scheme employed and its parameters are tuned with a particle swarm optimization (PSO) algorithm augmented with two chaotic maps for achieving an improved performance.
Abstract: This paper investigates the operation of a hybrid power system through a novel fuzzy control scheme. The hybrid power system employs various autonomous generation systems like wind turbine, solar photovoltaic, diesel engine, fuel-cell, aqua electrolyzer etc. Other energy storage devices like the battery, flywheel and ultra-capacitor are also present in the network. A novel fractional order (FO) fuzzy control scheme is employed and its parameters are tuned with a particle swarm optimization (PSO) algorithm augmented with two chaotic maps for achieving an improved performance. This FO fuzzy controller shows better performance over the classical PID, and the integer order fuzzy PID controller in both linear and nonlinear operating regimes. The FO fuzzy controller also shows stronger robustness properties against system parameter variation and rate constraint nonlinearity, than that with the other controller structures. The robustness is a highly desirable property in such a scenario since many components of the hybrid power system may be switched on/off or may run at lower/higher power output, at different time instants.

182 citations

### Cites background from "Functional Fractional Calculus"

• ...The robustness is a highly desirable property in such a scenario since many components of the hybrid power system may be switched on/off or may run at lower/higher power output, at different time instants....

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Journal ArticleDOI
TL;DR: In this article, a model for a power-counting renormalizable field theory living in a fractal spacetime has been proposed and the action is Lorentz covariant and equipped with a Stieltjes measure.
Abstract: We propose a model for a power-counting renormalizable field theory living in a fractal spacetime. The action is Lorentz covariant and equipped with a Stieltjes measure. The system flows, even in a classical sense, from an ultraviolet regime where spacetime has Hausdorff dimension 2 to an infrared limit coinciding with a standard D-dimensional field theory. We discuss the properties of a scalar field model at classical and quantum level. Classically, the field lives on a fractal which exchanges energy-momentum with the bulk of integer topological dimension D. Although an observer experiences dissipation, the total energy-momentum is conserved. The field spectrum is a continuum of massive modes. The gravitational sector and Einstein equations are discussed in detail, also on cosmological backgrounds. We find ultraviolet cosmological solutions and comment on their implications for the early universe.

154 citations

### Cites background from "Functional Fractional Calculus"

• ...J H E P 0 3 ( 2 0 1 0 ) 1 2 0 such as weather and stochastic financial models [34], to system modeling and control in engineering [35]....

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##### References
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Book ChapterDOI
01 Jan 2011
TL;DR: In this article, a series of applications where fractional calculus is finding application are described, starting with diffusion model in electrochemistry, electrode electrolyte interface, capacitor theory, fractance circuits, and application in feed back control systems, viscoelasticity, and vibration damping system.
Abstract: In this chapter a series of applications are described where fractional calculus is finding application. We start with diffusion model in electrochemistry, electrode electrolyte interface, capacitor theory, fractance circuits, and application in feed back control systems, viscoelasticity, and vibration damping system. This survey cannot cover complete applications like modern trends in electromagnetic theory like fractional multipole, hereditary prediction of gene behavior, fractional neural modeling in bio-sciences, communication channel traffic models, chaos theory, hence simple applications are provided for appreciation. However in the feedback control system section attempt is made to provide vector state feed back controller and observer available for multivariate control science, with explanation of fractional order feedback control and fractional phase shaper design to achieve robust iso-damped close loop performance.

4 citations