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Buchtitelbild

2017 | OriginalPaper | Buchkapitel

1. Essentials of Fractional Calculus

verfasst von : A. M. Mathai, H. J. Haubold

Erschienen in: Fractional and Multivariable Calculus

Verlag: Springer International Publishing

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Abstract

In recent decades, the field of fractional calculus has attracted interest of researchers in several areas including mathematics, physics, chemistry, engineering, and even finance and social sciences.

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Nächstes Kapitel Multivariable Calculus
Fußnoten
1
If f(x) is piecewise differentiable, then the formula (1.2.2) holds true at all points where f(x) is continuous and the integral in it must be understood in the sense of the Cauchy principal value.
 
2
A sufficient condition of the existence of the Laplace transform is that the original function is of exponential order as \(t \rightarrow \infty . \) This means that some constant \(a_f \) exists such that the product \( \hbox {e}^{-a _f t}\, |f(t)|\) is bounded for all t greater than some T. Then \(\widetilde{f}(s)\) exists and is analytic in the half plane \(\mathfrak {R}(s) > a_f. \) If f(t) is piecewise differentiable, then the formula (1.2.4) holds true at all points where f(t) is continuous and the (complex) integral in it must be understood in the sense of the Cauchy principal value.
 
3
For the existence of the Mellin transform and the validity of the inversion formula, we need to recall the following theorems TM1, TM2 adapted from Marichev’s [113] treatise, TM1 Let \(f(r) \in L^c(\epsilon ,E),\,0<\epsilon<E<\infty ,\) be continuous in the intervals \((0,\epsilon ],\,[E,\infty ),\) and let \(\,|f(r) | \le M\, r^{-{\gamma _1}}\) for \(0<r<\epsilon ,\) \(\,|f(r) | \le M\, r^{-{\gamma _2}}\) for \(r>E,\) where M is a constant. Then for the existence of a strip in the s-plane in which \(f(r)\, r^{s-1}\) belongs to \(L^c(0,\infty )\), it is sufficient that \(\gamma _1<\gamma _2. \) When this condition holds, the Mellin transform \(f^*(s)\) exists and is analytic in the vertical strip \(\gamma _1<\gamma =\mathfrak {R}(s) <\gamma _2. \) TM2 If f(t) is piecewise differentiable, and \(f(r)\, r^{\gamma -1} \in L^c(0, \infty ),\) then the formula (1.2.6) holds true at all points where f(r) is continuous and the (complex) integral in it must be understood in the sense of the Cauchy principal value.
 
4
We apply to Eq. (1.7.3) the fractional integral operator of order \(\beta \), namely \(\,_0I_t^\beta \). For \(\beta \in (0,1] \) we have:
$$\begin{aligned} _0I_t^\beta \,\circ \, _0^*D_t^{\beta }\, r(x,t)= \,_0I_t^\beta \,\circ \, _0I_t^{1-\beta }\, D_t^1\, r(x,t) = \,_0I_t^1\, D_t^1\, r(x,t) = r(x,t) - r(x,0^+)\,. \end{aligned}$$
For \(\beta \in (1,2] \) we have:
$$\begin{aligned} _0I_t^\beta \,\circ \, _0^*D_t^{\beta }\, r(x,t)\!=\! \,_0I_t^\beta \,\circ \, _0I_t^{2-\beta }\, D_t^2\, r(x,t)\! = \! \,_0I_t^2\, D_t^2\, r(x,t) \!=\! r(x,t) - r(x,0^+) -r_t(x,0^+). \end{aligned}$$
 
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Metadaten
Titel
Essentials of Fractional Calculus
verfasst von
A. M. Mathai
H. J. Haubold
Copyright-Jahr
2017
Buch
Fractional and Multivariable Calculus

Print ISBN: 978-3-319-59992-2

Electronic ISBN: 978-3-319-59993-9

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-59993-9_1

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