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2011 | Buch

Functional Fractional Calculus

verfasst von: Shantanu Das

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

When 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.

Inhaltsverzeichnis

Frontmatter

Introduction to Fractional Calculus

Introduction

Fractional calculus is three centuries old as the conventional calculus, but not very popular amongst science and or engineering community. The beauty of this subject is that fractional derivatives (and integrals) are not a local (or point) property (or quantity). Thereby this considers the history and non-local distributed effects. In other words perhaps this subject translates the reality of nature better! Therefore to make this subject available as popular subject to science and engineering community, adds another dimension to understand or describe basic nature in a better way. Perhaps fractional calculus is what nature understands and to talk with nature in this language is therefore efficient. For past three centuries this subject was with mathematicians and only in last few years, this is pulled to several (applied) fields of engineering and science and economics. However recent attempt is on to have definition of fractional derivative as local operator specifically to fractal science theory. Next decade will see several applications based on this three hundred years (old) new subject, which can be thought of as superset of fractional differintegral calculus, the conventional integer order calculus being a part of it. Differintegration is operator doing differentiation and sometimes integrations in a general sense.

Shantanu Das

Functions Used in Fractional Calculus

Introduction

This chapter presents a number of functions that have been found useful in the solution of the problems of fractional calculus. The base function is the Gamma function, which generalizes the factorial expression, used in multiple differentiation and repeated integrations, in integer order calculus. The Mittag-Leffler function is the basis function of fractional calculus, as exponential function is to integer order calculus. Several modifications of the Mittag-Leffler functions, along with other variants are introduced which are developed since 1903, for study of the fractional calculus. These functions are called Higher Transcendental Functions and its use in solving Fractional Differential Equations is as similar to use of transcendental functions for solving Integer Order Differential Equations. Use of these functions is demonstrated for solving Fractional differential equations with Laplace Transform Technique. Here, some interesting physical interpretation is given, for memory integrals for relaxation laws for generalized system dynamics (with memory); along with basic definition and physical interpretation of rough functions, and its fractal dimension. Several examples are solved to get fractional integration and fractional differentiation of standard function and use of introduced higher transcendental functions is demonstrated especially for solving Fractional Differential Equations.

Shantanu Das

Observation of Fractional Calculus in Physical System Description

Introduction

Fractional calculus allows a more compact representation and problem solution for some spatially distributed systems. Spatially distributed system representation allows a better understanding of the fractional calculus. The idea of fractional integrals and derivatives has been known since the development of regular calculus. Although not well known to most engineers, prominent mathematicians as well as scientists of the operational calculus have considered the fractional calculus. Unfortunately many of the results in the fractional calculus are given in language of advanced analysis and are not readily accessible to the general engineering and science community. Many systems are known to display fractional order dynamics. Probably the first physical system to be widely recognized as one demonstrating fractional behavior is the semi-infinite lossy (RC) transmission line. The current into the line is equal to the half-derivative of the applied voltage.

Shantanu Das

Concept of Fractional Divergence and Fractional Curl

Introduction

Fractional kinetic equations of the diffusion are useful approach for the description of transport dynamics in complex systems, which are governed by anomalous diffusion and non-exponential relaxation patterns. The anomalous diffusion can be modeled by fractional differential equation in time as well as space. For the spatial part use of fractional divergence modifies the anomalous diffusion expression, in the modified Fick’s law. Application of this fractional divergence is bought out in Nuclear reactor neutron flux definition. When anomalous diffusion is observed in time scale, the modification suggests use of Fractional kinetic equations. The evolution of Fractional Difference Equation, with reference to Fractional Brownian motion and the anomalous diffusion is also discussed in this chapter. Fractional curl operators will play perhaps role in electromagnetic theory and Maxwell equations. Here example in Electromagnetic is taken to have a feel how the fractional curl operator can map E and H fields in between the dual solutions of Maxwell equation.

Shantanu Das

Fractional Differintegrations Insight Concepts

Introduction

This chapter describes the geometric and physical interpretation of fractional integration and fractional differentiation. As a start point the Riemann-Liouville (RL) fractional integration is taken. Briefly existence of fractional differintegration is discussed along with useful tricks to obtain the fractional differintegration. The geometric interpretation is developed first for RL integration process along with concept of transformed time scales, and in-homogeneous time axis. Thereafter the RL definition is geometrically explained by convolution of the power function and the integrand, and as area under shape changing curve is demonstrated. The concept of delay is developed for Grunwald-Letnikov differintegration process and this is converted into the specific definition of short-memory principle, used for computer applications. The GL differintegration is also explained as in the classical calculus by considering infinitesimal quantities for the independent variable and the function, and explained graphically. The GL definition is expanded with binomial coefficients and its application to numerical regression. These methods are advance algorithms to get digital realization for fractional order controllers. The application to solve fractional differential equation numerically is demonstrated. Small introduction is made regarding definitions of Local Fractional Derivatives (LFD) for continuous but nowhere differentiable functions. These LFD (Kolwankar-Gangal K-G definition’s) utility is extended to measure critical point behaviors of physical system and its relation to ‘fractal’ dimension. The demonstration is made to have fractional integration and fractional differentiation, for fractal distributed quantities; thus, line, surface and volume integration can be performed when the measurable quantities are distributed in fractal form, Thereby generalizing the Gauss’s and Stroke’s law for fractal distributed quantities.

Shantanu Das

Initialized Differintegrals and Generalized Calculus

Introduction

This chapter demonstrates the need for a non-constant initialization for the fractional calculus. Here basic definitions are formed for the initialized fractional differintegrals (differentials and integrals). Here two basic popular definitions of fractional calculus are considered, those are Riemann-Liouville (RL) and Grunwald-Letnikov (GL). Two forms of initialization methods are prevalent, the ‘eterminal initialization’ and the ‘side initialization’. The issue of initialization has been an essentially a neglected subject in the development of the fractional calculus. Liouville’s choice of lower limit as –∞ and Riemann’s choice as c were in fact were issues related to the same initialization. Ross and Caputo maintained that to satisfy the composition of the fractional differintegrals, the integrated function and its integer order derivatives must be zero, for all times up to and including the start of fractional differintegration. Ross stated, “The greatest difficulty in Riemann’s theory is the interpretation of complimentary function. The question of existence of complimentary function caused much of confusion. Liouvelli was led to error and Riemann became inextricably entangled in his concept of a complimentary function.” The complimentary function issue is raised here because an initialization function, ‘which accounts for effect of history’, of the function, for fractional derivatives and integrals, will appear in the definitions of this chapter. The form of initialization function is kept similar to what Riemann has used as complimentary function ψ (x) however it’s meaning and use is different.

Shantanu Das

Generalized Laplace Transform for Fractional Differintegrals

Introduction

Differential equations of fractional order appear more and more frequently in various research areas of science and engineering. An effective method for solving such equations is needed. The method of Laplace transforms technique gives almost unified approach to solve the fractional differential equations. Also generalization of the same in view of initial conditions appropriately put (terminal/side charging) gives unified generalized approach. Also the fundamental fractional order differential equation concept is touched; its solution is the fundamental time response, whose combination provides solution to complicated systems. From this transfer function is constructed with fractional pole, which is the transfer function of the fundamental fractional differential equation, and is fundamental building block for more complicated fractional order systems. In this chapter scalar initialization and vector initialization problem is taken to describe the approaches developed for initialization function. These problems give insight into fractional “state” variable concepts and general system description of fractional order systems, and controls. For fractional order control system stability analysis transformed d Laplace s^q → w plane (wedge) is introduced. The pole-placement and its properties for control system stability for fractional order systems are carried on in this w-plane. The realization of fractional Laplace operator by rational function approximation is also introduced in this chapter. Here generalized stationary conditions are discussed and idea is developed as generalized Laplace transform to define Riemann-Liouvelli and/ or Caputo derivative (or even a derivative having the mix of two!).

Shantanu Das

Application of Generalized Fractional Calculus in Electrical Circuit Analysis and Electromagnetics

Introduction

The fractional calculus is widely popular, especially in the field of viscoelasticity. In this chapter variety of applications are discussed. This chapter is application oriented to demonstrate the fundamental of generalized (fractional) calculus developed earlier, with particular reference to initialization concepts. Here the treatment is to show coupling effect of the initialization functions, and the use of developed Laplace technique. The applications and potential applications of fractional calculus are in diffusion process, electrical science, electrochemistry, material creep, viscoelasticity, control science, electro magnetic theory and several more. In this chapter the fractal distribution effect of charges and its electromagnetic parameters is developed, based on representation of these fractal distributions by fractional differential elemental volume, surface or line. The generalizations of set of Maxwell’s equations are carried out for fractal distributions. The chapter also introduces the concept of representing fractional order transfer functions by rational polynomial ratios; also discusses issues of digitizing those fractional order transfer functions, and realize them through circuit techniques. This chapter restricted to electronics and electrical circuit models, and ‘fractal’ electromagnetic.

Shantanu Das

Application of Generalized Fractional Calculus in Other Science and Engineering Fields

Introduction

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.

Shantanu Das

System Order Identification and Control

Introduction

For unknown systems, ‘system identification’ has become the standard tool of the control engineer and scientists. Identifying a given system from data becomes more difficult, however when fractional orders are allowed. Here identification process is demonstrated using assumption that system order distribution is a continuous one. Frequency domain system identification can thus be performed using numerical methods demonstrated in this chapter. Here one concept of r-Laplace transforms is discussed (Laplace transform in log domain), to discuss the system order distribution. Also mentioned is variable order identification as further development where the system order also varies with ambient and time is highlighted. Here in this chapter, an identification method based on continuous order distribution, is discussed. This technique is suitable for both the standard integer order and fractional order systems. This is topic for further advance research as to qualify the procedure of system order identification and to have technique of tackling variable order. Extending this continuous order distribution discussion, the advance research of having a continuum order feedback and generalized PID control is elucidated. Also in this chapter some peculiarities of the pole property of fractional order system as ultra-damping, hyper-damping and fractional resonance is explained. Elaborate research in this direction is ongoing process; to crisply define the system identification, crisply define the variable order structure, along with generalized controller for future applications. The system identification in presence of disorder is what is challenging and some unification of disordered time-response that is relaxation is too discussed. This is general process of returning to equilibrium for say any stable system or properties of condense matter physics. The introduction to complex order calculus in system identification is too touched upon, in this chapter, along with identification of main parameters of ‘irregular’ stochastically behaving systems.

Shantanu Das

Solution of Generalized Differential Equation Systems

Introduction

Mathematical modeling of many engineering and physics problems leads to extraordinary differential equations (Non-linear, Delayed and Fractional Order). We call them Generalized Dynamic System. An effective method is required to analyze the mathematical model which provides solutions conforming to physical reality. For instance a Fractional Differential Equation (FDE), where the leading differential operator is Reiman-Liouvelli (RL) type requires fractional order initial states which are sometimes hard to physically relate. Therefore, we must be able to solve these dynamic systems, in space, time, frequency, area, and volume, with physical reality conserved. The usual procedures, like Runga-Kutta, Grunwarld-Letnikov Discretization with short memory principle etc, necessarily change the actual problems in essential ways in order to make it mathematically tractable by conventional methods. Unfortunately, these changes necessarily change the solution; therefore, they can deviate, sometimes seriously, from the actual physical behavior. The avoidance of these limitations so that physically correct solutions can be obtained would add in an important way to our insight into natural behavior of physical systems and would offer a potential for advances in science and technology. Adomian Decomposition Method (ADM) is applied here in this by physical process description; where a process reacts to external forcing function. This reactions-chain generates internal modes from zero mode reaction to first mode, second mode and to infinite modes; instantaneously in parallel time or space-scales; at the origin and the sum of all these modes gives entire system reaction. By this approach formulation of Fractional Differential Equation (FDE) by RL method it is found that there is no need to worry about the fractional initial states; instead one can use integer order initial states (the conventional ones) to arrive at solution of FDE. ADM method was first explored by mathematicians Prof Rasajit Bera and Prof S Saharay, for obtaining solutions to Fractional Order Differential equations.

Shantanu Das

Backmatter

Titel: Functional Fractional Calculus
verfasst von: Shantanu Das
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-20545-3
Print ISBN: 978-3-642-20544-6
DOI: https://doi.org/10.1007/978-3-642-20545-3

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