Initialized fractional calculus

In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus.

Composition rule of differintegral

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

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

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

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

Example

It is instructive to consider elementary integer-order calculus to understand the composition law. Let us first integrate then differentiate, using the example function $3x^{2}+1$:

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

Now, on exchanging the order of composition:

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

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 then differentiation (and vice versa) would not hold.

Description of initialization

Working with a properly initialized differintegral is the subject of initialized fractional calculus. If the differintegral 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 $\Psi$.

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

See also

• Initial conditions
• Dynamical systems

References

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