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**About this book :- **
**The Variable Order Fractional Calculus of Variations** written by
** Ricardo Almeida, Dina Tavares, Delfim Torres**.

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. Fractional calculus is a recent field of mathematical analysis, and it is a generalization of integer differential calculus, involving derivatives and integrals of real or complex order [17, 27]. The first note about this idea of differentiation, for non-integer numbers, dates back to 1695, with a famous correspondence between Leibniz and L’Hôpital. In a letter, L’Hôpital asked Leibniz about the possibility of the order n in the notation dny=dxn, for the nth derivative of the function y, to be a non-integer, n ¼ 1=2. Since then, several mathematicians investigated this approach, like Lacroix, Fourier, Liouville, Riemann, Letnikov, Grünwald, Caputo, and contributed to the grown development of this field. Currently, this is one of the most intensively developing areas of mathematical analysis as a result of its numerous applications.
The first book devoted to the fractional calculus was published by Oldham and Spanier in 1974, where the authors systematized the main ideas, methods, and applications about this field [18]. In the recent years, fractional calculus has attracted the attention of many mathematicians, but also some researchers in other areas like physics, chemistry, and engineering. As it is well known, several physical phenomena are often better described by fractional derivatives [13, 22, 30]. This is mainly due to the fact that fractional operators take into consideration the evolution of the system, by taking the global correlation, and not only local characteristics. Moreover, integer-order calculus sometimes contradict the experimental results, and therefore, derivatives of fractional order may be more suitable.

**Book Detail :- **
** Title: ** The Variable Order Fractional Calculus of Variations
** Edition: **
** Author(s): ** Ricardo Almeida, Dina Tavares, Delfim F. M. Torres
** Publisher: **
** Series: **
** Year: ** 2019
** Pages: ** 135
** Type: ** PDF
** Language: ** English
** ISBN: ** 978-3-319-94005-2, 978-3-319-94006-9
** Country: ** Portugal
** Get this book from Amazon**

**About Author :- **
** Ricardo Almeida **, Department of Mathematics, University of Aveiro, Aveiro, Portugal,
** Dina Tavares **, Polytechnic Institute of Leiria, Leiria, Portugal and
** Delfim F. M. Torres **, Department of Mathematics, University of Aveiro, Aveiro, Portugal.

**Book Contents :- **
**The Variable Order Fractional Calculus of Variations** written by
** Ricardo Almeida, Dina Tavares, Delfim Torres**
cover the following topics.
**1. Fractional Calculus**

1.1 Historical Perspective

1.2 Special Functions

1.3 Fractional Integrals and Derivatives

1.3.1 Classical Operators

1.3.2 Some Properties of the Caputo Derivative

1.3.3 Combined Caputo Derivative

1.3.4 Variable-Order Operators

1.3.5 Generalized Fractional Operators

1.3.6 Integration by Parts

References
**2. The Calculus of Variations **

2.1 The Classical Calculus of Variations

2.1.1 Euler–Lagrange Equations

2.1.2 Problems with Variable Endpoints

2.1.3 Constrained Variational Problems

2.2 Fractional Calculus of Variations

2.2.1 Fractional Euler–Lagrange Equations

2.2.2 Fractional Variational Problems of Variable-Order

References
**3. Expansion Formulas for Fractional Derivatives**

3.1 Caputo-Type Fractional Operators of Variable-Order

3.1.1 Caputo Derivatives for Functions of One Variable

3.1.2 Caputo Derivatives for Functions of Several Variables

3.2 Numerical Approximations

3.3 Example

3.4 Applications

3.4.1 A Time-Fractional Diffusion Equation

3.4.2 A Fractional Partial Differential Equation in Fluid Mechanics

References
**4. The Fractional Calculus of Variations**

4.1 Introduction

4.1.1 Combined Operators of Variable Order

4.1.2 Generalized Fractional Integration by Parts

4.2 Fundamental Variational Problem

4.2.1 Necessary Optimality Conditions

4.2.2 Particular Cases

4.2.3 Examples

4.3 Higher-Order Variational Problems

4.3.1 Necessary Optimality Conditions

4.3.2 Example

4.4 Variational Problems with Time Delay

4.4.1 Necessary Optimality Conditions

4.4.2 Example

4.5 Isoperimetric Problems

4.5.1 Necessary Optimality Conditions I

4.5.2 Necessary Optimality Conditions II

4.5.3 Example

4.6 Variational Problems with Holonomic Constraints

4.6.1 Necessary Optimality Conditions

4.6.2 Example

4.7 Fractional Variational Herglotz Problem

4.7.1 Fundamental Problem of Herglotz

4.7.2 Several Independent Variables

4.7.3 Examples

References

Appendix

Index

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