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(74)
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An Introduction to Fractional Calculus
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72212 Framework of causal distributions We could place ourselves within the framework of distributions, but convolution (which is the basic functional relations for invariant linear systems) is not, in general, associative When the supports are limited from below (usually the case when we are interested in causal signals), we obtain the property known as convolutive supports, which enables the associative convolution property Therefore, in D+ , which is a convolution algebra, there is an associative property of the convolution product, and the convolutive inverse of a distribution, if it exists, is unique (see lesson 32 of [GAS 90]), which allows a direct use of the impulse response h of a causal linear system Indeed, let us consider the general convolution equation in the unknown y D+ : P y=x (75)
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where P D+ represents the system and x D+ the known causal input; ie h, the impulse response of the system, de ned by: P h =
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Then, y = h x is the solution of equation (75); indeed: P y=P (h x) = (P h) x = x = x
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thanks to the associative convolution property of causal distributions We thus follow [GUE 72] to de ne fractional derivatives D DEFINITION 72 The derivative in the sense of causal distributions of f D+ is: D f = Y f
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where we have Y Y =
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(76)
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ie, Y is the convolutive inverse of Y in D+ At this stage, the problem is thus to identify the causal distribution Y , which we know could not be a function belonging to L1 Let us give its characterization by loc Laplace transform PROPOSITION 74 The Laplace transform of Y for > 0 is: L[Y ](s) = s for e(s) > 0 (77)
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ie, with a right-half complex plane as the convergence strip
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Proof We initially use the fact that, within the framework of causal distributions, the Laplace transform of a convolution product is the product of the Laplace transforms, which we apply to de nition Y using (76), by taking into account Proposition 72 and L[ ](s) = 1, ie: s L[Y ](s) = 1 s, e(s) > 0
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which proves, on the one hand, the existence and, on the other hand, the declared result NOTE 72 We read on the behavior at in nity of the Laplace transform that Y will be less regular the larger is PROPOSITION 75 The property Y Y = Y makes it possible to write the fundamental composition law: D D = D + for > 0 and > 0 PROPOSITION 76 For a causal distribution f which has a Laplace transform in e(s) > af , we have: L[D f ](s) = s L[f ](s) for e(s) > max(0, af ) (78)
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Finally, we obtain the following fundamental result PROPOSITION 77 For and two real numbers, we have: the property Y Y = Y + ; the fundamental composition law I I = I + ; by taking as notation convention I = D when < 0 EXAMPLE 72 We seek to clarify the half-order derivation: we rst calculate the distribution Y 1/2 , then we calculate D1/2 [f Y1 ] where f is a regular function From the point of view of distributions, we can write Y 1/2 = D1 Y1/2 , where D1 is the derivative in the sense of distributions; maybe, by taking C0 a test function: Y 1/2 , = D1 Y1/2 , = Y1/2 , = = = = 1 lim (1/2) 0 1 lim 2 (1/2) 0 1 (t) t
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An Introduction to Fractional Calculus
where pf indicates the nite part within the Hadamard concept of divergent integral We thus obtain the result, which is not very easy to handle in practice: Y 1/2 = pf (t+ ) ( 1/2)
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Let us now calculate the derivative of half-order of a causal distribution f Y1 where Y1 is the Heaviside distribution and f C 1 Then, we have D1 [f Y1 ] = f Y1 + f (0) , from where, by taking into account D1/2 = I 1/2 D1 : D1/2 [f Y1 ] = Y 1/2 [f Y1 ] = Y1/2 D1 [f Y1 ] = Y1/2 [f Y1 ] + f (0)Y1/2