The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag–Leffler kernel
- Arran Fernandez^{1}Email authorView ORCID ID profile and
- Dumitru Baleanu^{2, 3}
https://doi.org/10.1186/s13662-018-1543-9
© The Author(s) 2018
Received: 1 October 2017
Accepted: 5 March 2018
Published: 9 March 2018
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.
Keywords
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. There are several different definitions for derivatives and integrals (together referred to as differintegrals) in the fractional sense, which are classified in different categories. For example, the classical Riemann–Liouville and Caputo formulae are defined by integral transforms with power function kernels [1–4], while some more recent formulae [5–9] use integral transforms with various other kernel functions.
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 [13–15], drug release kinetics [16–19], 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.
Certain fundamental results of calculus have already been established in the AB model: Laplace transforms [7], integration by parts [27], the product rule and chain rule [26], etc. But as the idea is still so new, much remains to be done in this area. Furthermore, the AB model has found various applications, for example in chaos theory [28], variational calculus [27], and oscillators [29].
Specifically, our aim is to prove generalised versions of the mean value theorem and Taylor’s theorem in the AB model of fractional calculus. Analogous results are already known in the standard Riemann–Liouville [30] and Caputo [31] models, and versions of the mean value theorem for fractional difference operators have been proved in both the Caputo–Fabrizio model [32] and the AB model [33], but a fractional mean value theorem in the continuous AB model has not been established up until now. We shall also demonstrate some real-world applications of our results for modelling problems in fluid dynamics using a new fractional Boussinesq equation.
Our paper is structured as follows. In Sect. 2 we prove the main results and all required lemmas, and in Sect. 3 we redconsider some example Taylor expansions and discuss potential applications of our results.
2 Main results
2.1 The mean value theorem
The following result has been proved for example in [34], using Laplace transforms, and also in [26] using only the definition of AB derivatives and integrals.
Theorem 2.1
(AB Newton–Leibniz theorem)
We can use this fact to prove the following analogue of the mean value theorem for fractional derivatives in the AB model.
Theorem 2.2
(AB mean value theorem)
Proof
For interest’s sake we also include the following corollary, another form of the ABC fractional mean value theorem in terms of an inequality.
Corollary 2.1
Proof
2.2 Taylor’s theorem
Before starting to prove analogues of Taylor’s theorem for fractional AB derivatives, we first establish the following lemma.
Lemma 2.1
Proof
Now we are finally in a position to prove the following main result, our first analogue of Taylor’s theorem for fractional derivatives in the ABC model.
Theorem 2.3
(AB Taylor series about \(t=a\))
Proof
One disadvantage of Theorem 2.3 is that for many functions f, the ABC fractional derivative \({}^{\mathrm{ABC}}_{}D^{\alpha }_{a+}\,f(t)\) evaluated at the starting point \(t=a\) is zero. We can see this by considering the definition: since the ABC derivative is given by an integral from a to t, it will evaluate to zero given certain conditions on the behaviour of \(f(t)\) near \(t=a\). Thus, we present the following generalisation of Theorem 2.3, inspired by the work of [36].
Theorem 2.4
(AB Taylor series—general case)
Proof
We use formula (6) from Theorem 2.3 as our starting point, and apply it multiple times in different ways to derive (10).
Unfortunately, given the complexity of the formula for the remainder term \(R_{n+1}\), it will be difficult to tell whether and when series (10) converges as n goes to infinity. But we certainly have a valid finite series result, which can be verified computationally even for large values of n.
3 Examples and applications
As a basic example of the main result Theorem 2.4, let us consider what the series looks like with the particular function \(f(t)=(t-a)^{\beta}\).
Finally, we shall present an application of the new Taylor series given by Theorem 2.3.
The paper [37] used a fractional Taylor series for Caputo derivatives, namely the result of [31], to derive a new fractional Boussinesq equation, assuming a power law for the changes of flux in a control volume, as well as deriving a linear form of the same equation under an extra physical assumption. In the paper [38], this differential equation was used to model a water table profile between two parallel subsurface drains in both homogeneous and heterogeneous soils, and this application was verified by experiment.
4 Conclusions
During the last few years, a lot of attention was paid to modelling the dynamics of anomalous systems using fractional calculus. In our view, the best way is to start with fundamental principles appearing in nature, and after that to apply fractional techniques.
In this manuscript, we have proved the mean value theorem and Taylor’s theorem for derivatives defined in terms of a Mittag–Leffler kernel. Formulae (6) and (10) obtained for Taylor’s theorem in the ABC context appear different from classical and previous results, mainly due to the replacement of power functions with a more general form of summand.
These results can be used to model real-world problems such as the motion of unconfined groundwater, and we hope that they may find more such applications in the future.
Declarations
Acknowledgements
The first author is funded by a grant from the Engineering and Physical Sciences Research Council, UK.
Authors’ contributions
The first author contributed the results and proofs in Sect. 2, and the second author contributed the analysis in Sect. 3. The Introduction and Conclusions were joint efforts. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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