In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus.
The composition law of the differintegral operator states that although:
DqD-q=I
wherein D-q is the left inverse of Dq, the converse is not necessarily true:
D-qDq ≠ I
Consider elementary integer-order calculus. Below is an integration and differentiation using the example function
3x2+1
| d | |
| dx |
\left[\int(3x2+1)dx\right]=
| d | |
| dx |
[x3+x+C]=3x2+1,
Now, on exchanging the order of composition:
\int\left[
| d | |
| dx |
(3x2+1)\right]=\int6xdx=3x2+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
\Psi
| q | |
| D | |
| t |
f(t)=
| 1 | |
| \Gamma(n-q) |
| dn | |
| dtn |
| t | |
| \int | |
| 0 |
(t-\tau)n-q-1f(\tau)d\tau+\Psi(x)