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E-raamat: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications

(Technical University of Kosice, Slovak Republic)
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Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their ApplicationsSuurem pilt
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This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'.
This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more adequate mathematical models.
In this book, not only applied scientists, but also pure mathematicians will find fresh motivation for developing new methods and approaches in their fields of research.
A reader will find in this book everything necessary for the initial study and immediate application of fractional derivatives fractional differential equations, including several necessary special functions, basic theory of fractional differentiation, uniqueness and existence theorems, analytical numerical methods of solution of fractional differential equations, and many inspiring examples of applications.


Key Features
* A unique survey of many applications of fractional calculus
* Presents basic theory
* Includes a unified presentation of selected classical results, which are important for applications
* Provides many examples
* Contains a separate chapter of fractional order control systems, which opens new perspectives in control theory
* The first systematic consideration of Caputo's fractional derivative in comparison with other selected approaches
* Includes tables of fractional derivatives, which can be used for evaluation of all considered types of fractional derivatives

This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'.
This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more adequate mathematical models.
In this book, not only applied scientists, but also pure mathematicians will find fresh motivation for developing new methods and approaches in their fields of research.
A reader will find in this book everything necessary for the initial study and immediate application of fractional derivatives fractional differential equations, including several necessary special functions, basic theory of fractional differentiation, uniqueness and existence theorems, analytical numerical methods of solution of fractional differential equations, and many inspiring examples of applications.

Key Features
* A unique survey of many applications of fractional calculus
* Presents basic theory
* Includes a unified presentation of selected classical results, which are important for applications
* Provides many examples
* Contains a separate chapter of fractional order control systems, which opens new perspectives in control theory
* The first systematic consideration of Caputo's fractional derivative in comparison with other selected approaches
* Includes tables of fractional derivatives, which can be used for evaluation of all considered types of fractional derivatives

Arvustused

"...This is by no means the first (or the last) book on the subject of fractional calculus, but indeed it is one that wouldundoubtedly attract the attention (and successfully serve the needs) of mathematical, physical, and engineering scientists looking for applications of fractional calculus. I, therefore, recommend this well-written book to all users of fractional calculus." --H. M. Srivastava, Zentralblatt MATH

Preface xvii
Acknowledgments xxiii
Special Functions of the Fractional Calculus
1(40)
Gamma Function
1(15)
Definition of the Gamma Function
1(1)
Some Properties of the Gamma Function
2(2)
Limit Representation of the Gamma Function
4(2)
Beta Function
6(4)
Contour Integral Representation
10(2)
Contour Integral Representation of 1/Γ(z)
12(4)
Mittag-Leffler Function
16(21)
Definition and Relation to Some Other Functions
17(3)
The Laplace Transform of the Mittag-Leffler Function in Two Parameters
20(1)
Derivatives of the Mittag-Leffler Function
21(2)
Differential Equations for the Mittag-Leffler Function
23(1)
Summation Formulas
23(1)
Integration of the Mittag-Leffler Function
24(5)
Asymptotic Expansions
29(8)
Wright Function
37(4)
Definition
37(1)
Integral Representation
37(1)
Relation to Other Functions
38(3)
Fractional Derivatives and Integrals
41(80)
The Name of the Game
41(2)
Grunwald-Letnikov Fractional Derivatives
43(19)
Unification of Integer-order Derivatives and Integrals
43(5)
Integrals of Arbitrary Order
48(4)
Derivatives of Arbitrary Order
52(3)
Fractional Derivative of (t -- a)β
55(2)
Composition with Integer-order Derivatives
57(2)
Composition with Fractional Derivatives
59(3)
Riemann-Liouville Fractional Derivatives
62(15)
Unification of Integer-order Derivatives and Integrals
63(2)
Integrals of Arbitrary Order
65(3)
Derivatives of Arbitrary Order
68(4)
Fractional Derivative of (t -- a)β
72(1)
Composition with Integer-order Derivatives
73(1)
Composition with Fractional Derivatives
74(1)
Link to the Grunwald-Letnikov Approach
75(2)
Some Other Approaches
77(9)
Caputo's Fractional Derivative
78(3)
Generalized Functions Approach
81(5)
Sequential Fractional Derivatives
86(2)
Left and Right Fractional Derivatives
88(2)
Properties of Fractional Derivatives
90(13)
Linearity
90(1)
The Leibniz Rule for Fractional Derivatives
91(6)
Fractional Derivative of a Composite Function
97(1)
Riemann-Liouville Fractional Differentiation of an Integral Depending on a Parameter
98(1)
Behaviour near the Lower Terminal
99(2)
Behaviour far from the Lower Terminal
101(2)
Laplace Transforms of Fractional Derivatives
103(6)
Basic Facts on the Laplace Transform
103(1)
Laplace Transform of the Riemann-Liouville Fractional Derivative
104(2)
Laplace Transform of the Caputo Derivative
106(1)
Laplace Transform of the Grunwald-Letnikov Fractional Derivative
106(2)
Laplace Transform of the Miller-Ross Sequential Fractional Derivative
108(1)
Fourier Transforms of Fractional Derivatives
109(3)
Basic Facts on the Fourier Transform
109(1)
Fourier Transform of Fractional Integrals
110(1)
Fourier Transform of Fractional Derivatives
111(1)
Mellin Transform of Fractional Derivatives
112(9)
Basic Facts on the Mellin Transform
112(3)
Mellin Transform of the Riemann Liouville Fractional Integral
115(1)
Mellin Transform of the Riemann Liouville Fractional Derivative
115(1)
Mellin Transform of the Caputo Fractional Derivative
116(1)
Mellin Transform of the Miller-Ross Fractional Derivative
117(4)
Existence and Uniqueness Theorems
121(16)
Linear Fractional Differential Equations
122(4)
Fractional Differential Equation of a General Form
126(5)
Existence and Uniqueness Theorem as a Method of Solution
131(2)
Dependence of a Solution on Initial Conditions
133(4)
The Laplace Transform Method
137(12)
Standard Fractional Differential Equations
138(6)
Ordinary Linear Fractional Differential Equations
138(2)
Partial Linear Fractional Differential Equations
140(4)
Sequential Fractional Differential Equations
144(5)
Ordinary Linear Fractional Differential Equations
144(2)
Partial Linear Fractional Differential Equations
146(3)
Fractional Green's Function
149(10)
Definition and Some Properties
150(3)
Definition
150(1)
Properties
150(3)
One-term Equation
153(1)
Two-term Equation
154(1)
Three-term Equation
155(1)
Four-term Equation
156(1)
General Case: n-term Equation
157(2)
Other Methods for the Solution of Fractional-order Equations
159(40)
The Mellin Transform Method
159(2)
Power Series Method
161(7)
One-term Equation
162(4)
Equation with Non-constant Coefficients
166(1)
Two-term Non-linear Equation
167(1)
Babenko's Symbolic Calculus Method
168(5)
The Idea of the Method
169(1)
Application in Heat and Mass Transfer
170(2)
Link to the Laplace Transform Method
172(1)
Method of Orthogonal Polynomials
173(26)
The Idea of the Method
174(5)
General Scheme of the Method
179(2)
Riesz Fractional Potential
181(5)
Left Riemann-Liouville Fractional Integrals and Derivatives
186(2)
Other Spectral Relationships For the Left Riemann-Liouville Fractional Integrals
188(1)
Spectral Relationships For the Right Riemann-Liouville Fractional Integrals
189(2)
Solution of Arutyunyan's Equation in Creep Theory
191(1)
Solution of Abel's Equation
192(1)
Finite-part Integrals
192(3)
Jacobi Polynomials Orthogonal with Non-integrable Weight Function
195(4)
Numerical Evaluation of Fractional Derivatives
199(24)
Riemann-Liouville and Grunwald-Letnikov Definitions of the Fractional-order Derivative
199(1)
Approximation of Fractional Derivatives
200(3)
Fractional Difference Approach
200(1)
The Use of Quadrature Formulas
200(3)
The ``Short-Memory'' Principle
203(1)
Order of Approximation
204(4)
Computation of coefficients
208(1)
Higher-order approximations
209(1)
Calculation of Heat Load Intensity Change in Blast Furnace Walls
210(9)
Introduction to the Problem
211(1)
Fractional-order Differentiation and Integration
211(1)
Calculation of the Heat Flux by Fractional Order Derivatives - Method A
212(3)
Calculation of the Heat Flux Based on the Simulation of the Thermal Field of the Furnace Wall - Method B
215(3)
Comparison of the Methods
218(1)
Finite-part Integrals and Fractional Derivatives
219(4)
Evaluation of Finite-part Integrals Using Fractional Derivatives
220(1)
Evaluation of Fractional Derivatives Using Finite-part Integrals
220(3)
Numerical Solution of Fractional Differential Equations
223(20)
Initial Conditions: Which Problem to Solve?
223(1)
Numerical Solution
224(1)
Examples of Numerical Solutions
224(18)
Relaxation-oscillation Equation
224(1)
Equation with Constant Coefficients: Motion of an Immersed Plate
225(6)
Equation with Non-constant Coefficients: Solution of a Gas in a Fluid
231(4)
Non-Linear Problem: Cooling of a Semi-infinite Body by Radiation
235(7)
The ``Short-Memory'' Principle in Initial Value Problems for Fractional Differential Equations
242(1)
Fractional-order Systems and Controllers
243(18)
Fractional-order Systems and Fractional-order Controllers
244(7)
Fractional-order Control System
244(1)
Fractional-order Transfer Functions
245(1)
New Function of the Mittag-Leffler Type
246(1)
General Formula
247(1)
The Unit-impulse and Unit-step Response
248(1)
Some Special Cases
248(1)
PIλDμ-controller
249(1)
Open-loop System Response
250(1)
Closed-loop System Response
250(1)
Example
251(6)
Fractional-order Controlled System
252(1)
Integer-order Approximation
252(1)
Integer-order PD-controller
253(3)
Fractional-order Controller
256(1)
On Fractional-order System Identification
257(2)
Conclusion
259(2)
Survey of Applications of the Fractional Calculus
261(48)
Abel's Integral Equation
261(7)
General Remarks
262(1)
Some Equations Reducible to Abel's Equation
263(5)
Viscoelasticity
268(9)
Integer-order Models
268(3)
Fractional-order Models
271(4)
Approaches Related to the Fractional Calculus
275(2)
Bode's Analysis of Feedback Amplifiers
277(1)
Fractional Capacitor Theory
278(1)
Electrical Circuits
279(11)
Tree Fractance
279(1)
Chain Fractance
280(2)
Electrical Analogue Model of a Porous Dyke
282(1)
Westerlund's Generalized Voltage Divider
282(4)
Fractional-order Chua-Hartley System
286(4)
Electroanalytical Chemistry
290(1)
Electrode-Electrolyte Interface
291(2)
Fractional Multipoles
293(1)
Biology
294(2)
Electric Conductance of Biological Systems
294(1)
Fractional-order Model of Neurons
295(1)
Fractional Diffusion Equations
296(2)
Control Theory
298(1)
Fitting of Experimental Data
299(6)
Disadvantages of Classical Regression Models
299(1)
Fractional Derivative Approach
300(1)
Example: Wires at Nizna Slana Mines
301(4)
``Fractional-order'' Physics?
305(4)
Appendix: Tables of Fractional Derivatives 309(4)
Bibliography 313(24)
Index 337


Igor Podlubny is an Associate Professor at the Faculty of Mining, Ecology, Process Control, and Geotechnology of the Technical University of Kosice. He received his MSc in applied mathematics degree and Ph.D. degree in differential equations and mathematical physics from the Odessa State University, Ukraine, and the RNDr degree from the Comenius University in Bratislava, Slovakia. His work and interests focus on applications of mathematics in other fields, and especially on applications of differential equations of an arbitrary order.
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