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Fractional Differential Equation
Fractional differential equations (FDE) are natural generalizations of differential equations through the application of fractional calculus. FDE can have one or more fractional derivatives in it. For example, simplest fractional differential equation can be written as


 * $$ _aD_t^\alpha y(t)= c $$

where $$ \textstyle _aD_t^\alpha $$ is any fractional derivative and $$ c $$ is a constant. The solution of this FDE is given by


 * $$ y(t)= ...$$

A slightly general, FDE can be given by


 * $$ _aD_t^\alpha y(t)= \lambda y(t)$$ with the initial condition $$ \big[_aD_t^\alpha y(t)\big]_0 =b_0 $$

The corresponding solution given by


 * $$ y(t)= b_0 E_\alpha(\lambda t)$$

where $$ E_\alpha(t)$$ is the Mittag-Leffler function of the $$ \textstyle 1^{st}$$ kind.