Initialized fractional calculus
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In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus.
Composition rule of differintegral[edit]
A certain counterintuitive property of the differintegral operator should be pointed out, namely the composition law. Although
wherein D−q is the left inverse of Dq, the converse is not necessarily true:
Example[edit]
It is instructive to consider elementary integer-order calculus to see what's happening. First, integrate then differentiate, using the example function 3x2 + 1:
on exchanging the order of composition:
In which the constant of integration is c. Even if it wasn't 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[edit]
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, not just a constant or set of constants. This is called complementary function .
Working with a properly initialized differintegral is the subject of initialized fractional calculus.
See also[edit]
References[edit]
- Lorenzo, Carl F.; Hartley, Tom T. (2000), Initialized Fractional Calculus (PDF), NASA (technical report).