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About this book

​The Variable-Order Fractional Calculus of Variations is devoted to the study of fractional operators with variable order and, in particular, variational problems involving variable-order operators. This brief presents a new numerical tool for the solution of differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided.
The contributors consider variational problems that may be subject to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, they establish necessary optimality conditions of Euler–Lagrange type. As the terminal point in the cost integral is free, as is the terminal state, transversality conditions are also obtained.
The Variable-Order Fractional Calculus of Variations is a valuable source of information for researchers in mathematics, physics, engineering, control and optimization; it provides both analytical and numerical methods to deal with variational problems. It is also of interest to academics and postgraduates in these fields, as it solves multiple variational problems subject to one or more constraints in a single brief.

Table of Contents

Frontmatter

Chapter 1. Fractional Calculus

Abstract
In this chapter, a brief introduction to the theory of fractional calculus is presented. We start with a historical perspective of the theory, with a strong connection with the development of classical calculus (Sect. 1.1). Then, in Sect. 1.2, we review some definitions and properties about a few special functions that will be needed. We end with a review on fractional integrals and fractional derivatives of noninteger order and with some formulas of integration by parts, involving fractional operators (Sect. 1.3).
Ricardo Almeida, Dina Tavares, Delfim F. M. Torres

Chapter 2. The Calculus of Variations

Abstract
As part of this book is devoted to the fractional calculus of variations, in this chapter, we introduce the basic concepts about the classical calculus of variations and the fractional calculus of variations. The study of fractional problems of the calculus of variations and respective Euler–Lagrange-type equations is a subject of current strong research.
Ricardo Almeida, Dina Tavares, Delfim F. M. Torres

Chapter 3. Expansion Formulas for Fractional Derivatives

Abstract
In this chapter, we present a new numerical tool to solve differential equations involving three types of Caputo derivatives of fractional variable-order. For each one of them, an approximation formula is obtained, which is written in terms of standard (integer order) derivatives only. Estimations for the error of the approximations are also provided. Then, we compare the numerical approximation of some test function with its exact fractional derivative. We present an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable-order.
Ricardo Almeida, Dina Tavares, Delfim F. M. Torres

Chapter 4. The Fractional Calculus of Variations

Abstract
In this chapter, we consider general fractional problems of the calculus of variations, where the Lagrangian depends on a combined Caputo fractional derivative of variable fractional order \(^CD_\gamma ^{{\alpha (\cdot ,\cdot )},{\beta (\cdot ,\cdot )}}\) given as a combination of the left and the right Caputo fractional derivatives of orders, respectively, \({\alpha (\cdot ,\cdot )}\) and \({\beta (\cdot ,\cdot )}\). More specifically, here we study some problems of the calculus of variations with integrands depending on the independent variable t, an arbitrary function x and a fractional derivative \(^CD_\gamma ^{{\alpha (\cdot ,\cdot )},{\beta (\cdot ,\cdot )}}x\).
Ricardo Almeida, Dina Tavares, Delfim F. M. Torres

Backmatter

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