Introduction to Fractional Calculus#

Author:: Pavan Pranjivan Mehta, SISSA, International School of Advanced Studies, Italy

Introduction#

Fractional calculus is a generalised form of the integer-order calculus. While an integer-order derivative is a local operator, a fractional derivative is a non-local operator. The notion of Brownian motion is extended to admit Levy stable processes in the case of fractional diffusion. Many different operators have been described as fractional derivatives and integrals, with different properties and behaviours, and there are also further generalisations within non-local calculus. Real-world applications of non-local models can be found in turbulence .

In view of the importance of the subject, in the forthcoming sections, we define the fractional derivatives and their properties

Grünwald-Letnikov Definition#

In order to derive the Grünwald-Letnikov fractional derivative , consider,

\[f'(x) = {d f \over d x} = \lim_{h \rightarrow 0} {f(x) - f(x-h) \over h}\]

\[f''(x) = {d^{2} f \over d x^{2}} = \lim_{h \rightarrow 0} {f(x) - 2f(x-h) + f(x-2h) \over h^{2}}\]

\[f'''(x) = {d^{3} f \over d x^{3}} = \lim_{h \rightarrow 0} {f(x) - 3 f(x-h) + 3 f(x-2h) - f(x-3h) \over h^{3}}\]

therefore, for a $p^{th}$-order differentiation, where $p \in \mathbb{N}$ (and $f \in C^{p}[a, b]$), we have,

\[\label{eq:2.1} f^{p}(x) = {d^{p} f \over d x^{p}} = \lim_{h \rightarrow 0} {1 \over h^{p}} \sum_{r=0}^{p} (-1)^{r} \binom{p}{r} f (x - r h)\]

where,

\[\binom{p}{r} = {p (p -1) (p-2) (p-3) \dots (p-r+1) \over r !}\]

Formula ([eq:2.1]) derived for $p \in \mathbb{N}$, can be equivalently re-written as ([eq:2.2]), by constructing a uniform grid $x \in [a, b]$, with $N = (x-a)/h$. Since the terms after $\binom{p}{p}$, vanishes . Although, formula ([eq:2.2]) is derived for $p \in \mathbb{N}$, but it is valid for any non-integer order ($p > 0$). Thus, the Grünwald-Letnikov fractional derivative is defined as ([eq:2.2]) for $p \in \mathbb{R}_+$,

\[\label{eq:2.2} f^{p}(x) = {d^{p} f \over d x^{p}} = \lim_{h \rightarrow 0} {1 \over h^{p}} \sum_{r=0}^{N} (-1)^{r} \binom{p}{r} f (x - r h)\]

However, for non-integer order the terms after $\binom{p}{p}$ does not vanish. Thus Diethelm in imposed an additional condition on the function ($f$) as ([eq:2.3]), its equivalency is argued in theorem 2.25 of .

\[\begin{split}\label{eq:2.3} f(x)=\begin{cases} f(x), & \text{if $x \in [a, b]$}.\\ 0, & \text{otherwise}. \end{cases}\end{split}\]

For negative order of ($p$), consider ([eq:2.4]),

\[\label{eq:2.4} f^{-p}(x) = {d^{-p} f \over d x^{-p}} = \lim_{h \rightarrow 0} {1 \over h^{-p}} \sum_{r=0}^{N} (-1)^{r} \binom{-p}{r} f (x - r h)\]

where,

\[\begin{split}\binom{-p}{r} = {-p (-p -1) (-p-2) (-p-3) \dots (-p-r+1) \over r !} = (-1)^r {\left[\begin{matrix} ~p \\ ~r \end{matrix}\right]}\end{split}\]

The introduced notation ${\left[\begin{matrix} ~p \\ ~r \end{matrix}\right]}$, implies (following ),

\[\begin{split}{\left[\begin{matrix} ~p \\ ~r \end{matrix}\right]} = {p (p +1) (p+2) (p+3) \dots (p+r-1) \over r !}\end{split}\]

Thus, the Grünwald-Letnikov fractional integral is defined as ([eq:2.5]),

\[\begin{split}\label{eq:2.5} f^{-p}(x) = {d^{-p} f \over d x^{-p}} = \lim_{h \rightarrow 0} {h^{p}} \sum_{r=0}^{N} {\left[\begin{matrix} ~p \\ ~r \end{matrix}\right]} f (x - r h)\end{split}\]

Formula ([eq:2.2]) and ([eq:2.5]) is the unification of fractional derivative and integration, respectively, for arbitrary order ($p \in \mathbb{R}$)

Riemann-Liouville Definition#

In the section, we give the definition of the Riemann-Liouville operators. Recall the Cauchy’s formula for repeated integration ([eq:2.6]) for $p \in \mathbb{N}$,

\[\label{eq:2.6} \underbrace{ \int_{a}^{x} \int_{a}^{x_p} \int_{a}^{x_{p-1}} \dots \int_{a}^{x_{2}}}_{\text{p-integrals}} f(x_1) d x_1 d x_2 \dots d x_p = {1 \over (p-1)!} \int_{a}^{x} {(x - \tau)}^{p-1} f (\tau) d \tau\]

Here, $p \in \mathbb{N}$, to generalise this formula ([eq:2.6]) introduce $\Gamma(p) = (p-1)!$, where $\Gamma(.)$ is the Euler gamma function, thus the Riemann-Liouville fractional integral is defined as ([eq:2.7]) for ($p \in \mathbb{R}_+$)

\[\label{eq:2.7} {}_a I^p_x f(x) ~:=~ {1 \over \Gamma(p)} \int_{a}^{x} {(x - \tau)}^{p-1} f (\tau) d \tau\]

Note the notation introduced to denote Riemann-Liouville fractional integral (${}_a I^p_x$), which implies $p-th$ integration performed, with the lower limit as $a$ and upper limit as $x$. Theorem 2.1 of gives the existence proof. Further, we state the theorem 2.2 of , the semi-group property of Riemann-Liouville fractional integrals as ([eq:2.8]) for $p, q \in \mathbb{R}_+$,

\[\label{eq:2.8} {}_a I^p_x ~ {}_a I^q_x ~ f(x) ~=~ {}_a I^{p+q}_x ~ f(x)\]

Subsequently, the Riemann-Liouville fractional derivative is defined as ([eq:2.9]) for $p \in \mathbb{R}_+$ and $k \in \mathbb{N}$,

\[\label{eq:2.9} {}_a^{RL} D^p_x f(x) ~:=~ {1 \over \Gamma(k-p)} {d^k \over d x^k} \int_{a}^{x} {(x - \tau)}^{k-p-1} f (\tau) d \tau ~~, ~ k-1 \leq p < k\]

here, ($d^k / dx^k$) is the classical integer-order derivative. The Riemann-Liouville fractional derivative in terms of the Riemann-Liouville fractional integral is denoted as ([eq:2.10]),

\[\label{eq:2.10} {}_a^{RL} D^p_x f(x) ~=~ {d^k \over d x^k} ~ {}_a I^{k-p}_x f(x) ~~, ~ k-1 \leq p < k\]

Lemma 2.12 of gives the existence proof for Riemann-Liouville fractional derivative. Indeed for $p \in \mathbb{N}$ the Riemann-Liouville fractional derivative is the classical integer-order derivative. Further, we state the theorem 2.13 of , the semi-group property of Riemann-Liouville fractional derivative as ([eq:2.11]) for $p, q \in \mathbb{R}_+$,

\[\label{eq:2.11} {}_a^{RL} D^p_x ~ {}_a^{RL} D^q_x ~f(x) ~=~ {}_a^{RL} D^{p+q}_x ~ f(x)\]

We further state the theorem 2.14 of as ([eq:2.12]),

\[\label{eq:2.12} {}_a^{RL} D^p_x ~ {}_a I^p_x ~f(x) ~=~ f(x)\]

Upon performing the limit in ([eq:2.2]), it can be shown that the Grünwald-Letnikov fractional derivative coincides with Riemann-Liouville fractional derivative. Thus the two definitions are equivalent to each other (refer section 2.3.7 of and section 3.3 of )

Caputo Definition#

Although, the Riemann-Liouville fractional derivative is mathematically well established, there are two key problems for application to physical systems (refer section 2.4 of ).

For a constant function ($f$), the ${}_a^{RL} D^p_x ~f \neq 0$, in general. It is zero only for a special as $a \rightarrow -\infty$. This is not very convenient for application to physical systems.
The initial conditions are specified as $\lim_{t \rightarrow a} ~{}_a^{RL} D^p_t ~f(t) = g$. Such a initial condition does not make physical sense, since we measure the function value explicitly as $f(t) |_{t=0} = g$ (refer section 2.4 of its problem with initial condition is further elaborated by taking the Laplace transform)

As a remedy, Caputo defined a fractional derivative as ([eq:2.13]) for $p \in \mathbb{R}_+$ and $k \in \mathbb{N}$,

\[\label{eq:2.13} {}_a^{C} D^p_x f(x) ~:=~ {1 \over \Gamma(k-p)} \int_{a}^{x} {(x - \tau)}^{k-p-1} {d^k \over d x^k} f (\tau) d \tau ~~, ~ k-1 < p \leq k\]

Indeed, the Caputo fractional derivative of a constant is zero and the initial conditions are defined in a classical way. Theorem 3.1 of , expresses the Caputo fractional derivative in terms of Riemann-Liouville fractional derivative by introduction of a Taylor polynomial. We further state theorem 3.7 of as ([eq:2.14]) ,

\[\label{eq:2.14} {}_a^{C} D^p_x ~ {}_a I^p_x ~f(x) ~=~ f(x)\]

For homogeneous conditions the Caputo fractional derivative equivalent to the Riemann-Liouville fractional derivative. This can be shown by taking repeated integration by parts.

The semi-group property of Caputo fractional derivative is given as ([eq:2.14_1]) for $p, q > 0$, where $p, q$ can be integers too (refer lemma 3.13 of ).

\[\label{eq:2.14_1} {}_a^{C} D^p_x ~~ {}_a^{C} D^q_x ~f(x) ~=~ {}_a^{C} D^{p+q}_x ~f(x)\]

Left and right sided fractional derivative#

Upon close observation to the definitions introduced in the preceding section, it is evident that the fractional derivative only takes into account the non-local interactions from the left, hence there are termed as the left fractional derivative. A right fractional derivative can be defined too, which takes account of the non-local interactions from the right, which is a also a well defined object. Thus the mathematical theory of fractional calculus is complete.

In this paper, we employ both the left and right definitions; as physical process are amalgamation of interactions from both left and right boundary.

Again, we state the left Riemann-Liouville fractional derivative as ([eq:2.15]) for a function defined over an interval $[a,b]$, $p \in \mathbb{R}_+$ and $k \in \mathbb{N}$,

\[\label{eq:2.15} {}_a^{RL} D^p_x f(x) ~:=~ {1 \over \Gamma(k-p)} {d^k \over d x^k} \int_{a}^{x} {(x - \tau)}^{k-p-1} f (\tau) d \tau ~~, ~ k-1 \leq p < k\]

while, the right Riemann-Liouville fractional derivative is defined as ([eq:2.16]) for $p \in \mathbb{R}_+$ and $k \in \mathbb{N}$, notice the domain of integration.

\[\label{eq:2.16} {}_x^{RL} D^p_b f(x) ~:=~ { (-1)^k \over \Gamma(k-p)} {d^k \over d x^k} \int_{x}^{b} {(\tau - x)}^{k-p-1} f (\tau) d \tau ~~, ~ k-1 \leq p < k\]

Similarly, we state the left Caputo fractional derivative as ([eq:2.17]) for a function defined over an interval $[a,b]$, $p \in \mathbb{R}_+$ and $k \in \mathbb{N}$,

\[\label{eq:2.17} {}_a^{C} D^p_x f(x) ~:=~ {1 \over \Gamma(k-p)} \int_{a}^{x} {(x - \tau)}^{k-p-1} {d^k \over d x^k} f (\tau) d \tau ~~, ~ k-1 < p \leq k\]

while, the right Caputo fractional derivative is defined as ([eq:2.18]) for $p \in \mathbb{R}_+$ and $k \in \mathbb{N}$, again notice the domain of integration.

\[\label{eq:2.18} {}_x^{C} D^p_b f(x) ~:=~ { (-1)^k \over \Gamma(k-p)} \int_{x}^{b} {(\tau - x)}^{k-p-1} {d^k \over d x^k} f (\tau) d \tau ~~, ~ k-1 < p \leq k\]

The left and right definitions allow us to explicitly define the Riesz fractional derivative ([eq:2.19]) for a function defined over an interval $[a,b]$ and $c_p = - 0.5 cos (p \pi / 2)$,

\[\label{eq:2.19} {}_a^{RZ} D^p_b f(x) ~:=~ c_{p} \left ( {}_a^{RL} D^p_x f(x) ~+~ {}_x^{RL} D^p_b f(x) \right )\]

References#

[1] Pavan Pranjivan Mehta. Fractional and tempered fractional models for reynolds-averaged navier–stokes equations. , 24(11-12):507–553, 2023.

[2] Pavan Pranjivan Mehta, Guofei Pang, Fangying Song, and George Em Karniadakis. Discovering a universal variable-order fractional model for turbulent couette flow using a physics-informed neural network. , 22(6):1675–1688, 2019.

[3] I Podlubny. . Academic, 1999.

[4] Kai Diethelm. . Springer-Verlag, Berlin, 2010.