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 Dq is the left inverse of Dq, the converse is not necessarily true:


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 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[edit]

This is the problem that with the differintegral. 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 .

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

See also[edit]


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