Initialized fractional calculus

A certain counterintuitive property of the differintegral operator should be pointed out, namely the composition law. Although

${\displaystyle \mathbb {D} ^{q}\mathbb {D} ^{-q}=\mathbb {I} }$

wherein D^−q is the left inverse of D^q, the converse is not necessarily true:

${\displaystyle \mathbb {D} ^{-q}\mathbb {D} ^{q}\neq \mathbb {I} }$

This is the problem that with the differintegral. If the differintegral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, we lose information, as we lost the c in the first equation.

In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complementary function is needed, not just a constant or set of constants. We call this complementary function ${\displaystyle \Psi }$.

${\displaystyle \mathbb {D} _{t}^{q}f(t)={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{0}^{t}(t-\tau )^{n-q-1}f(\tau )\,d\tau +\Psi (x)}$

Working with a properly initialized differintegral is the subject of initialized fractional calculus.

• Initial conditions
• Dynamical systems

• Lorenzo, Carl F.; Hartley, Tom T. (2000), Initialized Fractional Calculus (PDF), NASA (technical report).