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The Variable-Order Fractional Calculus of Variations
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.Comment: 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, 201
Fractional Calculus in Wave Propagation Problems
Fractional calculus, in allowing integrals and derivatives of any positive
order (the term "fractional" kept only for historical reasons), can be
considered a branch of mathematical physics which mainly deals with
integro-differential equations, where integrals are of convolution form with
weakly singular kernels of power law type. In recent decades fractional
calculus has won more and more interest in applications in several fields of
applied sciences. In this lecture we devote our attention to wave propagation
problems in linear viscoelastic media. Our purpose is to outline the role of
fractional calculus in providing simplest evolution processes which are
intermediate between diffusion and wave propagation. The present treatment
mainly reflects the research activity and style of the author in the related
scientific areas during the last decades.Comment: 33 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1008.134
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