The Variable-Order Fractional Calculus of Variations

Abstract

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional calculus of variations and some fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler-Lagrange equations for several types of fractional problems of the calculus of variations, with or without constraints.

The final authenticated version of this preprint is available online as a SpringerBrief in Applied Sciences and Technology at [https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos, detected by the authors while reading the galley proofs, were corrected, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 2019

References in corpus (9)

• Caputo derivatives of fractional variable order: numerical approximations
• Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
• Fractional variational calculus of variable order
• Approximation of fractional integrals by means of derivatives
• Optimality Conditions for Fractional Variational Problems with Dependence on a Combined Caputo Derivative of Variable Order
• Combined Fractional Variational Problems of Variable Order and Some Computational Aspects
• Folded potentials in cluster physics - a comparison of Yukawa and Coulomb potentials with Riesz fractional integrals
• Constrained fractional variational problems of variable order
• Fractional variational calculus in terms of a combined Caputo derivative

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• Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection
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• Combined Fractional Variational Problems of Variable Order and Some Computational Aspects
• Fractional-Order Modelling and Optimal Control of Cholera Transmission
• Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets
• Fractional Scalar Field Cosmology
• A new spectral method based on two classes of hat functions for solving systems of fractional differential equations and an application to respiratory syncytial virus infection
• On a Non-Newtonian Calculus of Variations
• A Stochastic Fractional Calculus with Applications to Variational Principles
• Time-Fractional Optimal Control of Initial Value Problems on Time Scales
• On time-fractional partial differential equations of time-dependent piecewise constant order
• Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems
• Analysis of fractional integro-differential equations of thermistor type
• A survey on fractional variational calculus
• Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces
• The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only
• Weak Pontryagin's Maximum Principle for Optimal Control Problems Involving a General Analytic Kernel
• Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials
• A Necessary Optimality Condition for Extended Weighted Generalized Fractional Optimal Control Problems
• Existence Results for a Multipoint Fractional Boundary Value Problem in the Fractional Derivative Banach Space