Abstract
We establish analogues of the mean value theorem and Taylor’s theorem for fractional differential operators defined using a Mittag–Leffler kernel. We formulate a new model for the fractional Boussinesq equation by using this new Taylor series expansion.
Highlights
1 Introduction The importance of fractional calculus, i.e. the study of differentiation and integration to non-integer orders, started to be appreciated during the last few decades, mainly because many successful models were developed in various branches of science and engineering
The classical Riemann–Liouville and Caputo formulae are defined by integral transforms with power function kernels [1,2,3,4], while some more recent formulae [5,6,7,8,9] use integral transforms with various other kernel functions
4 Conclusions During the last few years, a lot of attention was paid to modelling the dynamics of anomalous systems using fractional calculus
Summary
The importance of fractional calculus, i.e. the study of differentiation and integration to non-integer orders, started to be appreciated during the last few decades, mainly because many successful models were developed in various branches of science and engineering. Fractional derivatives and integrals have found many applications across a huge variety of fields of science—for example in financial models [10], geohydrology [11], chaotic systems [12], epidemiology [13,14,15], drug release kinetics [16,17,18,19], nuclear dynamics [20], viscoelasticity [21], complexity theory [22], bioengineering [23], image processing [24], and so on. One of the reasons for their broad usefulness is their non-locality: ordinary derivatives are local operators, while fractional ones (at least according to most definitions) are non-local, having some degree of memory. For this reason, they are often useful in problems involving global optimisation, such as those appearing in control theory
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