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Essentials of Fractional Calculus

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Book cover Fractional and Multivariable Calculus

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 122))

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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.

This chapter is based on the lectures by Professor Francesco Mainardi of the University of Bologna, Italy.

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Notes

  1. 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. 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. 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. 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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    A. M. Mathai & H. J. Haubold

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    A. M. Mathai

  3. Vienna International Centre, Office for Outer Space Affairs, United Nations, Vienna, Austria

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Mathai, A.M., Haubold, H.J. (2017). Essentials of Fractional Calculus. In: Fractional and Multivariable Calculus . Springer Optimization and Its Applications, vol 122. Springer, Cham. https://doi.org/10.1007/978-3-319-59993-9_1

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