where (s) = s + s is the propagation constant and the global re ection coef cient

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An Introduction to Fractional Calculus

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Figure 79 Position in the Laplace plane of the poles of the models with losses ( = 025) and without loss ( = 0) Legend: o: = 1, : = +08

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7531 Decomposition into wavetrains This is undoubtedly the most physically meaningful decomposition By expanding the fraction in (748) into power series, which is legitimate for e(s) > 0 in the case | | < 1, then by applying the inverse Laplace transform term by term, we obtain the following wavetrain decomposition of the system:

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where (t) is the fundamental solution of a 3D diffusion process (ie, parabolic heat equation): (t) = L 1 e

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In Figure 710a, we represent function 1 , that is, the elementary lossy wave: it is a function of class C of which all the derivatives are zero at t = 0+ and which decreases like t 3/2 at in nity; it is integrable and it even belongs to L1 , L2 , , L , 1 with norm p = 2(1 p ) 1 p An interesting property is that we again nd the case without loss of the classical wave equation: 0 (t) = (t) (as a limit in the sense of distributions), which does not have any of the regularity properties of when > 0 Moreover, the family of functions obeys the following scaling law: t (t) = 2 1 2

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02 04 06 08 1 t 12 14 16 18 2

05 t

Figure 710 (a) Fundamental solution 1 of a 3D diffusion process; (b) scaling law of functions for = 025 (continuous line), 075 (dotted line) and 125 (indents)

10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0

0 0 5 15

Figure 711 Impulse response of the model with losses: (a) for = 001, (b) for = 025

which is shown in Figure 710b Thus, it is clear that the waves which appear successively in h(t) in decomposition (749) have an increasingly low amplitude, but a temporal support (or a width with middle height) which is increasingly high We illustrate this phenomenon as follows In Figure 711a, = 001 is very small and the supports of successive waves remain separate To some extent this resembles the case without loss, but with successive amplitudes which decrease in a way nearly independent of the total re ection coef cient , whereas in Figure 711b, = 025 can no longer be compared with a perturbation, while the increasing spreading of the supports is veri ed 7532 Quasi-modal decomposition To apply the inverse Laplace transform to (748) directly by using residue calculus, we must take into account the cut along the negative real axis, which is imposed by the multiform character of s s

An Introduction to Fractional Calculus

It is this which very precisely creates an integral term in the modal decomposition, sometimes called aperiodic multimode, since we can regard it as the superposition of a continuous in nity of damped exponentials; the structure of the impulse response is thus the following: h(t) =

n S

c esn t + n

( ) e t d

(752)

where some of s possibly disappeared from the discrete sum on the indices in S n (this occurs very exactly when equation (747) does not have solutions such that e( s) > 0) In the diffusive part, = o( ) shows that in a certain way, the integral term is a perturbation of order NOTE 710 It should be noted that we observe here a generalization of the decomposition result (739), the only difference being the in nitely countable character of the poles For a detailed study of the family of diffusive representation ( ) indexed by , see 9 of [HEL 00] 7533 Fractional modal decomposition In (748), we carry out the decomposition of the meromorphic function in a series of normally convergent elementary meromorphic functions on every compact subset of the complex plane (see 5 of [CART 61]), by using: ( 1)n 1 = sinh(z) n= z in which gives us: 1 ( 1)n H(s) = 2 n= (s) + i(n + 0 ) Lastly, we break up each term of the series into rational fractions of the variable s: 1 1 1 1 = + + s + s s0 n n s n s n n