Page 11 - Mathematical Techniques of Fractional Order Systems

Chapter 1

Variable Order Fractional

Derivatives and Bone
Remodeling in the Presence of

Metastases

´
Joana P. Neto, Duarte Valerio and Susana Vinga
Universidade de Lisboa, Lisbon, Portugal

1.1 INTRODUCTION
This chapter deals with variable order fractional derivatives, which are a gen-
eralization of fractional order derivatives.

1.1.1 What Are Variable Order Fractional Derivatives?
Fractional order derivatives are, of course, themselves a generalization of usual
integer order derivatives and integrals, known from Calculus. Differentiation
and integration notions of orders nAN are, in Fractional Calculus, generalized
to account for orders αAR (Wheeler, 1997). But since all real numbers are now
available as orders, it is now rather easy to conceive variable order derivatives,
with an order that changes with time. Among the first papers on the subject
were Samko (1995) and Lorenzo and Hartley (2002a), mostly based upon the
Riemann Liouville definition of fractional derivatives. Variable order deriva-
tives have been further developed in Vale ´rio and Sa ´ da Costa (2011b); Vale ´rio
and Sa ´ da Costa (2013), using not only the Riemann Liouville definition,
but also the Gru ¨nwald Letnikoff and the Caputo definitions. Additional
insights are provided in Sierociuk et al. (2015a,b) and Sierociuk and
Malesza (2017), including experiments with electronic circuits imple-
menting the relevant mathematical operators.
These variable order derivatives are useful in practice, just as fractional
derivatives of constant order are. In fact, being more than 300 years old,

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00001-5
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