Initialized fractional calculus

In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus, a branch of mathematics dealing with derivatives of non-integer order, emerged nearly simultaneously with traditional calculus. This emergence was partly due to Leibniz's notation for derivatives of integer order: ${\frac {d^{n}}{dx^{n}}}$. Thanks to this notation, L'Hopital was able to inquire in a letter to Leibniz about the interpretation of taking $n={\frac {1}{2}}$ in a derivative. At that moment, Leibniz couldn't provide a physical or geometric interpretation for this question, so he simply replied to L'Hopital in a letter that "... is an apparent paradox from which, one day, useful consequences will be drawn" .

The name "fractional calculus" originates from a historical question, as this branch of mathematical analysis studies derivatives and integrals of a certain order $\alpha \in \mathbb {R}$. Currently, fractional calculus lacks a unified definition of what constitutes a fractional derivative. Consequently, when the explicit form of a fractional derivative is unnecessary, it is typically denoted as follows:

${\frac {d^{\alpha }}{dx^{\alpha }}}.$

Composition rule of Differintegrals

The composition law of the differintegral operator states that 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

Consider elementary integer-order calculus. Below is an integration and differentiation 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, and differentiation (and vice versa) would not hold.

Description of initialization

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 $\Psi$.

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

See also

• Initial conditions
• Dynamical systems

References

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