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Initialized fractional calculus

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In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus, a branch of mathematics dealing with derivatives of non-integer order.

Composition rule of Differintegrals

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The composition law of the differintegral operator states that although:

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

Example

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Consider elementary integer-order calculus. Below is an integration and differentiation using the example function :

Now, on exchanging the order of composition:

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

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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 .

See also

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References

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