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While stepping into the realm of fractional calculus, I have become confident on the RL fractional integral, defined as: $$^{RL}_aI^p_tf(x) = _p\int^t_af(x)dx^p = \frac{1}{\Gamma(p)}\int^t_a(t-x)^{p-1}f(x)dx$$ Where $a$ and $t$ are the integration limits and $p$ is the (fractional) integration order.
We can of course also use this to compute derivatives. This is pretty straightforward and easy to use. However, the other main way I have researched to do fractional differentiation is through Caputo's fractional derivative, which I have not found a solid definition for.
For context, I am a 16 year old maths student working by myself as I am interested in further calculus. I am struggling with reading alot of the research papers that come with higher education, and not the usual teaching websites. The best I have been able to find is: $$^C_aD^p_tf(x)=\frac{1}{\Gamma(n-p)}\int^x_a\frac{f^n(t)}{(x-t)^{p-n+1}}dt$$ Where $a$ and $t$ are the integration limits and $p$ is the (fractional) integration order once again.
My first question is what raising our function in the integrand to $n$ in $f^n(t)$ entails. My second question is if this equation is even correct, as every source seems to have it in a different format. When I tried to compute a basic half derivative with this it seemed to dissolve into an RL integral. I can't yet see the difference between these two methods, even when I keep getting assured they are very different ways to solve a problem!
Apologies if I have completely missed major points here, studying this level of mathematics is brand new but very interesting for me. Any book recommendations on this subject would be greatly appreciated too! Many thanks.

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  • $\begingroup$ Welcome! I did some digging and found this huge body of related work, presumably you can find a 'reliable' definition there with some background and applications. $\endgroup$
    – FShrike
    Oct 27, 2022 at 16:02

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The main difference appears to be in the order of differentiation.

See page 2 of this:

The problem regarding the “right” fractional derivatives is more delicate and has no unique solution. Presently, the main approach for introducing the fractional derivatives is to define them as the left-inverse operators to the fractional integrals. However, even for the Riemann-Liouville fractional integral, there exist infinitely many different families of operators that fulfill this property. In particular, for $0<\alpha\le1$, the Riemann-Liouville fractional derivative: $$\left(\mathfrak{D}_{\mathrm{RL}}^\alpha f\right)(t)=\frac{\mathrm{d}}{\mathrm{d}t}\left(\mathcal{I}_{0+}^{1-\alpha}f\right)(t)$$

The Caputo fractional derivative: $$\left(\mathfrak{D}_{\mathrm{C}}^\alpha f\right)(t)=\left(\mathcal{I}_{0+}^{1-\alpha}\frac{\mathrm{d}f}{\mathrm{d}\tau}\right)(t)$$

Where the author defines:

$$\left(\mathcal{I}_{a+}^\alpha f\right)(t)=\frac{1}{\Gamma(\alpha)}\int_a^t(t-\tau)^{\alpha-1}f(\tau)\,\mathrm{d}\tau$$

This is backed up by this which writes:

In his 1967 paper, Caputo reformulated the definition of the Riemann–Liouville fractional derivative, by switching the order of the ordinary derivative with the fractional integral operator. By doing so, the Laplace transform of this new derivative depends on integer order initial conditions, differently from the initial conditions when we use the Riemann–Liouville fractional derivative, which involve fractional order conditions. Motivated by this concept, we present the following definition.

This hints at one difference in how these derivatives might be applied to fractional differential equations. According to both of these sources, it seems that the 'right' Caputo derivative should be, for some $\varphi>0$, $\varphi=n+\alpha$, $0\le alpha<1$:

$$\mathfrak{D}^{\varphi}_{\mathrm{C}}(f)(t)=\frac{\mathrm{d}^n}{\mathrm{d}t^n}\left[\frac{1}{\Gamma(1-\alpha)}\int_0^t(t-\tau)^{-\alpha}f'(\tau)\,\mathrm{d}\tau\right]$$

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  • $\begingroup$ Many thanks, this has cleared so much up. It looks like the "Laplace transform" is important for this so I will look further into that. As you said the main difference seems to be the order of differentiation and integration. Your source is much easier to understand than others, so anybody else on this topic in future should definetly visit link. Truly interesting stuff! $\endgroup$
    – B-Foster16
    Oct 27, 2022 at 22:46
  • $\begingroup$ @SixteenHundredX You’re very welcome. All I know about Laplace transforms is that they are a key tool in differential equation theory, and all I know about fractional calculus is that they’re useful in obscure modelling problems involving fractional differential equations. I can only guess the two go hand in hand, somehow. I too was quite interested in them a while back. Have you seen this? $\endgroup$
    – FShrike
    Oct 27, 2022 at 22:50
  • $\begingroup$ That's a great thread, and the real-world uses of fractional calculus are definetly mainly in differential equations. I have seen a few articles of links to some complex physics questions, but if this proves useful is a question for the future. At the moment the pure mathematical connections work so well it makes this topic very satisfying to work with. $\endgroup$
    – B-Foster16
    Oct 27, 2022 at 23:20

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