By Boling Guo, Xueke Pu, Fenghui Huang
This e-book goals to introduce a few new developments and effects at the learn of the fractional differential equations, and to supply a great knowing of this box to newcomers who're attracted to this box, that's the authors' appealing wish. This e-book describes theoretical and numerical elements of the fractional partial differential equations, together with the authors' researches during this box, equivalent to the fractional Nonlinear Schrödinger equations, fractional Landau Lifshitz equations and fractional Ginzburg Landau equations. It additionally covers sufficient primary wisdom at the fractional derivatives and fractional integrals, and sufficient historical past of the fractional PDEs.
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Extra info for Fractional Partial Differential Equations and Their Numerical Solutions
Obviously, K1 (t, τ ) = 1. Eqn. & Their Numerical Solu. Fractional integrals and derivatives t 35 τ dτ t f (τ1 )dτ1 = 0 frac-1-2 t f (τ )dτ 0 dτ1 τ 0 t (t − τ )f (τ )dτ, = 0 thus K1 (t, τ ) = (t − τ ). When n = 3, t τ dτ 0 τ1 dτ1 t f (τ2 )dτ2 = 0 τ (τ − τ1 )f (τ1 )dτ1 dτ 0 0 0 t = t (τ1 − τ )dτ1 f (τ )dτ τ 0 t = f (τ ) 0 (t − τ )2 dτ, 2 hence K2 (t, τ ) = (t − τ ) /2. Generally, Kn (t, τ ) = (t − τ )n−1 /(n − 1)! 1) 0 where Γ(n) = (n− 1)!. Assume f ∈ C[0, T ], the space of continuous functions on [0, T ], then for arbitrary t ∈ [0, T ], the integral exists in the sense of Riemann integral for any n 1.
15) k=0 We can show that if f ∈ C , then for arbitrary l = 0, 1, 2, · · · , there holds D l [D μ−m−k f (t)] = Dl+μ−m−k f (t). Hence, if f ∈ C , n k Cμ−m Dm [Dk tn ][D −m+μ−k f (t)] μ n D [t f (t)] = k=0 n m k Cμ−m = k=0 j Cm [D j+k tn ][D μ−j−k f (t)]. Eqn. & Their Numerical Solu. 16) Cμr [D r tn ][Dμ−r f (t)], = μ > 0. 7 Suppose that f ∈ C and g is analytical at t for arbitrary t ∈ [0, T ]. Then for any ν > 0 and 0 < t T , there holds ∞ D μ [f (t)g(t)] = Cνk [D k g(t)][D −ν−k f (t)]. 5, and hence omitted.
Also, some existence results of fractional ordinary equations are obtained by iteration. For readers’ convenience, some basics of Fourier transform, Laplace transform and Mittag-Leﬄer function are given at the end of the chapter. 1 Fractional integrals and derivatives Riemann-Liouville fractional integrals To introduce R-L fractional integral, consider ﬁrst the following iteration integrals t D−1 [f ](t) = f (τ )dτ, 0 t D −2 [f ](t) = τ1 dτ1 0 f (τ )dτ, 0 ··· t D−n [f ](t) = τ1 dτ1 0 τn−1 dτ2 · · · 0 f (τ )dτ.
Fractional Partial Differential Equations and Their Numerical Solutions by Boling Guo, Xueke Pu, Fenghui Huang
Categories: Mathematical Physics