分数阶微分方程文献综述.pdf

1 (Fractional calculus), , . . 1695, (L’Hˆopital) (Leibniz) n 1 “D f , n = 2 , ?”, . , N.H.AbelM.CaputoL.Euler J.FourierA.K.Gru¨unwald J.Hadamard G.H.HardyP.S.LaplaceG.W.LeibnizA.V.LetnikovJ.Liouvil-leB.Riemann . , , , ; , . : 1 , , ; 2 ; 3 , . 30, . , . , , “ ” “ ” , , , . OldhamSpanier[1]MillerRoss[2] Podlubny[3] Hilfer[4] Kilbas, SrivastavaTrujillo[5] Sabatier, AgrawalTenreiro Machado[6] , . , , . 2 n n n D f = d f/dx , f , n , f n . , n , , , “”, n , . , . . N.H.Abel (Euvres comple`tes de Niels Henrik Abel, vol.1, Grondahl, Christiania, 1881, pp. 11-18) α 1 x ′ s (η)dη α = (x), 0 (x − η) 1 d−α (x) s(x) = −α . Γ(1 − α) dx , , . . 2.1 , , , . DelboscoRodino[7] (Dα y)(x) = f (x,y(x)); y(k) = yk ∈ R(k = 0, 1, , [α]), 0+ 0 ≤ x ≤ T, α 0, f (x,y)[0, 1] × R . , Schauder, [0,δ](0 δ ≤ 1)y(x). , f [x,y(x)]Lipschitz, , C [0, 1]. . , Hayek et al.[8] (Dα n 0+y)(x) = A(x)y(x) + B (x), y(a) = b(0 α ≤ 1; a 0; b ∈ R ) . Lipschitzian, Diethelm[9] C [0,h] . ,