Initialized fractional calculus

The composition law of the differintegral operator states that 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:

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

Consider elementary integer-order calculus. Below is an integration and differentiation using the example function ${\displaystyle 3x^{2}+1}$:

${\displaystyle {\frac {d}{dx}}\left[\int (3x^{2}+1)dx\right]={\frac {d}{dx}}[x^{3}+x+C]=3x^{2}+1\,,}$

Now, on exchanging the order of composition:

${\displaystyle \int \left[{\frac {d}{dx}}(3x^{2}+1)\right]=\int 6x\,dx=3x^{2}+C\,,}$

Where C is the constant of integration. Even if it was not obvious, the initialized condition ƒ'(0) = C, ƒ''(0) = D, etc. could be used. If we neglected those initialization terms, the last equation would show the composition of integration, and differentiation (and vice versa) would not hold.

Working with a properly initialized differ integral is the subject of initialized fractional calculus. If the differ integral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, information is lost, as with C in the first equation.

However, in fractional calculus, given that the operator has been fractionalized and is thus continuous, an entire complementary function is needed. This is called complementary function ${\displaystyle \Psi }$.

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

• Initial conditions
• Dynamical systems