An Introduction to Fractional Calculus

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starting equation, to check the coherence of the result Second, the question of initial conditions for fractional differential equations is not truly solved: we are obliged to de ne zero or in nite initial values Lastly, the true analytical nature of solutions can be masked by closed-form solutions utilizing a great number of special functions, which facilitates neither the characterization of important analytical properties of these solutions nor their numerical simulation The focus of our work concerns the theory of fractional differential equations (FDE): rst, we clarify various de nitions by using the framework of causal distributions (ie, generalized functions whose support is the positive real axis) and by interpreting results on functions expandable in fractional power series of order ( -FPSE); second, we clarify problems related to fractional differential equations by formulating solutions in a compact general form and third, we establish a strong bond with diffusive representations of pseudo-differential operators (DR of PDO), which is a nearly incontrovertible concept when derivation orders are arbitrary Finally, we study the extension to several variables by treating a fractional partial differential equation (FPDE) which in fact constitutes a modal analysis of fractional order 713 Outline This chapter is composed of four distinct sections First, in section 72, we give de nitions of the fundamental concepts necessary for the study and handling of fractional formalism We recall the de nition of fractional integration in section 721 We show in section 722 that the inversion of this functional relation can be correctly de ned within the framework of causal distributions and we examine the fundamental solutions directly connected to this operator Lastly, we adopt a de nition which is easier to handle, ie, a mild fractional derivative, so as to be able to use fractional derivatives on regular causal functions We examine in section 723 the eigenfunctions of this new operator and show its structural relationship with a generalization of Taylor expansions for non-differentiable functions at the temporal origin, like the functions expandable in fractional power series Then, in section 73, we are interested in the fractional differential equations These are linear relations in an operator of fractional derivative and its successive powers; it appears naturally that the rational orders play an important role, since certain powers are in direct relationship with the usual derivatives of integer orders We thus examine fractional differential equations in the context of causal distributions (in section 732) and functions expandable into fractional power series (in section 733) We then tackle, in section 734, the asymptotic behavior of the fundamental solutions of these fractional differential equations, ie, the divergence in modulus, the pseudo-periodicity or convergence towards zero of the eigenfunctions

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Scaling, Fractals and Wavelets

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of fractional derivatives (which plays a similar role to that of the exponential function in the case of integer order) Finally, in section 735, we examine a class of controlled-and-observed linear dynamic systems of fractional order and approach some typical stakes of automatic control Then, in section 74, we consider fractional differential equations in one variable but when orders of derivations are not commensurate: there are no simple algebraic tools at our disposal in the frequency domain and work carried out in the case of commensurate orders does not apply any more We further examine the strong bond which exists with diffusive representations of pseudo-differential operators We give some simple ideas and elementary properties and then we present a general result of decomposition for the solutions of fractional differential equations into a localized or integer order part and a diffusive part In section 75, nally, we show that the preceding theory in the time variable (which appeals, in the commensurate case, to polynomials and rational fractions in frequency domain) can extend to several variables in the case of fractional partial differential equations (we obtain more general meromorphic functions which are not rational fractions) With this intention, we treat an example conclusively: that of the partial differential wave equation with viscothermal losses at the walls of the acoustic pipes, ie an equation which reveals a time derivative of order three halves Throughout this chapter, we will treat the half-order as an example, in order to clarify our intention This chapter has been inspired by several articles and particularly [AUD 00, MAT 95a] This personal work is also the fruit of collaborations with various researchers including d Andr a-Novel, Audounet, Dauphin, Heleschewitz and Montseny More recently, new co-authors have helped enlarge the perspective of our work: let us mention H lie, Haddar, Prieur and Zwart 72 De nitions 721 Fractional integration The primitive, canceling at initial time t = 0, reiterated an integer number n of times I n f , of an integrable function f , is nothing other than the convolution of f with a polynomial kernel Yn (t) = tn 1 /(n 1)!: + I n f (t) =

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