Initialized fractional calculus

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

## Contents

* 1 Composition rule of differintegral
* 1.1 Example
* 2 Description of initialization
* 3 See also
* 4 References

## Composition rule of differintegral

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

𝔻q𝔻−q=𝕀

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

𝔻−q𝔻q≠𝕀

### Example

It is instructive to consider elementary integer-order calculus to see what's happening. First, integrate then differentiate, using the example function 3x2 + 1:

ddx[∫(3x2+1)dx]=ddx[x3+x+c]=3x2+1,

on exchanging the order of composition:

∫[ddx(3x2+1)]=∫6xdx=3x2+c,

in which the constant of integration is c. Even if it wasn't obvious, the initialization terms 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

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 Ψ.

𝔻tqf(t)=1Γ(n−q)dndtn∫0t(t−τ)n−q−1f(τ)dτ+Ψ(x)

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

## See also

* Initial conditions
* Dynamical systems

## References

* Lorenzo, Carl F.; Hartley, Tom T. (2000), Initialized Fractional Calculus, NASA, https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20000031631.pdf (technical report).

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