FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 25We will consider various versions of the equation (1.1). For γ 0 = γ 1 = 1,γ 3 = γ 4 = · · · = γ n = 0, the equation (1.1) turns into the equation∫1xΓ(1 − γ 2 )0u ′′ (t)(x − t) γ dt − [λ + q(x)]u(x) = 0, (1.3)2which is called a fractional oscillating equation [5], and the operator D (σ 2)is called the operator of fractional differentiation in Caputo sense [5]. Forγ 0 = γ 2 = 1, γ 3 = γ 4 = · · · = γ n = 0, the equation (1.1) turns into theequation1Γ(1 − γ 1 )∫dxdx0u ′ (t)(x − t) γ dt − [λ + q(x)]u(x) = 0. (1.4)1The equation (1.4) has been investigated as a model equation of fractionalorder 1 < σ < 2 (see [3] and references therein). Further, if γ 0 = γ 2 = · · · =γ n = 1, then the equation (1.1) will be written asD (σ n) u =1Γ(1 − γ 1 )d n−1dx n−1∫x0d(t)(x − t) γ 1 dt − [λ + q(x)]u(x) = 0. (1′ )The two-point boundary value problem of Dirichlet u(0) = 0, u(1) = 0for the fractional oscillatory equation was studied by one of the authorsof this paper in [5]. Therein for the first time Green’s function for similarboundary value problems has been constructed. In particular, it was provedthat the two-point problem of Dirichlet u(0) = 0, u(1) = 0 for the fractionaloscillatory equation with q(x) = 0 is equivalent to the equationu(x) =[ ∫xλ(1 − t) 1−γ 2u(t) dt −Γ(1 − γ 2 )0∫ 10]x 1−γ 2(1 − t) 1−γ 2u(t) dt .The same problem for the model fractional differential equation of order1 < σ < 2 is equivalent to the equation [3]u(x) =[ ∫xλ(1 − t) γ 1u(t) dt −Γ(1 + γ 1 )0∫ 10]x(1 − t) γ 1u(t) dt .The operator A inverse to the operator B induced by the differential expression(1 ′ ) and natural boundary conditionsu(0) = 0; D (σ 1) u ∣ ∣x=0= 0, . . . , D (σ n−2) u ∣ ∣x=0= 0, u(1) = 0looks like (see [3] and [6])[ ∫x1Au =Γ(ρ −1 x(1 − t) 1 ρ −1 u(t) dt −)0∫ 10]x 1 ρ −1 (1 − t) 1 ρ −1 u(t) dt .