Everything involving fractional calculus (FC) differential equations: in practical application of FC differential equations; in pure mathematics; numerical methods for the resolution of FDE; dynamical systems; fractional partial differential equations; FDE of distributed order; boundary value problems for FDE; techniques for studying FDE; in a variety of subjects ranging from mathematical physics to probability theory.
See fractional-calculus for more information about the underlying operations of fractional differentiation. This includes not only differential equations of non-intiger order, but also differential equations of functional order, e.g. the $x$-th intgeral of $y$ ($\operatorname{I}^{x}\left( y\left( x \right) \right) = x^{m}$, $m \in C$) wich would have a solution given by $y\left( x \right) = \operatorname{D}^{x}\left( x \right) = \frac{\Gamma\left( m + 1 \right)}{\Gamma\left( m - x + 1 \right)} \cdot x^{m - x}$ where $\Gamma\left( \cdot \right)$ is the Gamma Function which follows from Euler's attempt to define a definition of a Fractional Derivative for $x^{m}$.
Such operations might be applied either in a univariate setting, so that the tag ordinary-differential-equations would be useful, or in a multivariate setting, when the tag partial-differential-equations would be relevant.
A common approach to defining fractional derivatives in the univariate case involves integral transforms like Fourier or Mellin. Other constructions arise for fractional derivatives in the multivariate case to give trace theorems for Sobolev spaces and their generalizations.
One of the better known and solved FDEs is $$ \begin{align*} \operatorname{D}^{2 \cdot v} \left( y\left( x \right) \right) + a \cdot \operatorname{D}^{v} \left( y\left( x \right) \right) + b \cdot y\left( x \right) &= 0,\\ \end{align*} $$ which has a solution given by $$ \begin{align*} y\left( x \right) &= \begin{cases} \sum\limits_{k = 1}^{\frac{1}{v} - 1}\left( a^{\frac{1}{v} - k - 1} \cdot \operatorname{E}_{x}\left( -k \cdot v;\, a^{\frac{1}{v}} \right) - b^{\frac{1}{v} - k - 1} \cdot \operatorname{E}_{x}\left( -k \cdot v;\, b^{\frac{1}{v}} \right) \right), &\text{if } a \ne b\\ x \cdot \exp\left( a \cdot x \right) \cdot \sum\limits_{k = -\frac{1}{v} + 1}^{\frac{1}{v} - 1}\left( a^{k} \cdot \left( \frac{1}{v} - \left| k \right| \right) \cdot \exp\left( a^{\frac{1}{v}} \cdot x \right) \right), &\text{if } a = b \ne 0\\ \frac{1}{\Gamma\left( 2 \cdot v \right)} \cdot x^{2 \cdot v - 1 }, &\text{if } a = b = 0\\ \end{cases}, \end{align*} $$ where $\operatorname{E}_{t}\left( \cdot;\, \cdot \right)$ is the $\operatorname{E}_{t}$ Function.
