By R. Sneider

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**Best mathematical physics books**

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ICM 2002 satellite tv for pc convention on Nonlinear research was once held within the interval: August 14 18, 2002 at Taiyuan, Shanxi Province, China. This convention used to be geared up by means of Mathematical tuition of Peking collage, Academy of arithmetic and procedure Sciences of chinese language Academy of Sciences, Mathematical institution of Nankai college, and division of arithmetic of Shanxi collage, and was once backed by means of Shanxi Province schooling Committee, Tian Yuan arithmetic origin, and Shanxi college.

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3 The fractional derivative in terms of ﬁnite diﬀerences the Gr¨ unwald-Letnikov derivative Extending a general rule from n ∈ N to α ∈ C is a very helpful recipe to establish another deﬁnition of a fractional derivative in terms of a limit of ﬁnite diﬀerences. 50) h→0 ∞ = exp(kx) lim 1 − h→0 (−1)j j=0 ∞ = exp(kx) lim − h→0 (−1)j hj−1 j=1 = exp(kx) lim k − h h→0 (hk)j j! kj j! k3 k2 + h2 ... 2! 3! 54) which nicely coincides with the Liouville deﬁnition of a fractional derivative. 45) we have presented a ﬁrst unique deﬁnition of a fractional derivative, which is valid for any analytic function, as long as the series converges.

2 Mittag-Leﬄer functions Besides the gamma function Euler has brought to light an additional important function, the exponential: ∞ ez = zn n! 85 X α Fig. 2 Solutions and zeroes of the Caputo-wave equation are the fractional pendant of the trigonometric functions and special cases of the Mittag-Leﬄer function cos(α, x) = E2α (−x2α ) and the generalized Mittag-Leﬄer function sin(α, x) = xα E2α,1+α (−x2α ). The graph is given for diﬀerent α near α = 1. Units are given as multiples of π/2. 14) where we have introduced an arbitrary real number α > 0.

All these observed phenomena have diﬀerent physical causes. Within the framework of Newton’s theory they are summarized as friction forces FR . To be a little bit more speciﬁc we consider as friction forces all kinds of forces which point in the opposite direction of the velocity of a particle. Therefore an ansatz for friction forces is a simple power law: FR = −μ sign(v)|v|α with an arbitrarily chosen real exponent α. 8) November 8, 2013 17:18 BC: 8934 - Fractional Calculus HerrmannFC2˙main Friction Forces 35 α ≈ 0 is observed for static and kinetic friction for solids α = 1 Stokes friction in liquids with high viscosity α = 2 is a general trend for high velocities In reality both gases and liquids show a behavior which only approximately corresponds to these special cases.