Editing Fractional calculus (section)

===Caputo fractional derivative===
{{main|Caputo fractional derivative}}
Another option for computing fractional derivatives is the [[Caputo fractional derivative]]. It was introduced by [[Michele Caputo]] in his 1967 paper.<ref>{{cite journal |last=Caputo |first=Michele |title=Linear model of dissipation whose ''Q'' is almost frequency independent. II |journal=Geophysical Journal International |year=1967 |volume=13 |issue=5 |pages=529–539 |doi=10.1111/j.1365-246x.1967.tb02303.x |bibcode=1967GeoJ...13..529C |doi-access=free}}.</ref> In contrast to the Riemann–Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows, where again {{math|1=''n'' = ⌈''α''⌉}}:
<math display="block">\sideset{^C}{_t^\alpha}D f(t)=\frac{1}{\Gamma(n-\alpha)} \int_0^t \frac{f^{(n)}(\tau)}{\left(t-\tau\right)^{\alpha+1-n}}\, d\tau.</math>

There is the Caputo fractional derivative defined as:
<math display="block">D^\nu f(t)=\frac{1}{\Gamma(n-\nu)} \int_0^t (t-u)^{(n-\nu-1)}f^{(n)}(u)\, du \qquad (n-1)<\nu<n</math>
which has the advantage that it is zero when {{math|''f''(''t'')}} is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative. Moreover, there is the Caputo fractional derivative of distributed order defined as
<math display="block">\begin{align}
\sideset{_a^b}{^nu}Df(t) &= \int_a^b \phi(\nu)\left[D^{(\nu)}f(t)\right]\,d\nu \\
   &= \int_a^b\left[\frac{\phi(\nu)}{\Gamma(1-\nu)}\int_0^t \left(t-u\right)^{-\nu}f'(u)\,du \right]\,d\nu
\end{align}</math>

where {{math|''ϕ''(''ν'')}} is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.