FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

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CHAPTER 2The Sturm–Liouville Problems for a SecondOrder Ordinary Differential Equation withFractional Derivatives in the Lower Terms1. On Some Problems from the Theory of Equations of MixedType Leading to Boundary Problems for the DifferentialEquations of the Second Order with Fractional DerivativesConsider the equation⎧m∑⎪⎨ u ′′ + a 0 (x)u ′ + a i (x)D αi0x ω i(x)u + u m+1 (x)u = f(x),i=1(2.1)u(0) cos α + u⎪⎩′ (0) sin α = 0,u(1) cos β + u ′ (1) sin β = 0,where 0 < α m < · · · < α 1 < 1, D α is the operator of fractional differentiationof order αD σ u =1Γ(1 − α)∫dxdx0u(t)dt, 0 < σ < 1.(x − t)σMany direct and inverse problems associated with a degenerating hyperbolicequation and equation of the mixed hyperbolic-parabolic type arereduced to equations of the type (2.1). In [3] it is shown that to a problem(2.1) reduces an analogue of a problem of Tricomi type for the hyperbolicparabolicequation with Gellerstendt operator in the body.On Euclid planes with the cartesian orthogonal coordinates x and ywe will consider the model equation in partial derivatives of the mixed(parabolic-hyperbolic) type|y| mH(−1) ∂2 u∂x 2 = ∂1+H(−y) u, (2.2)∂y1+H(−y) where m = const > 0, H(y) is the Heavyside function, u = u(x, y). Theequation (2.1) in the upper half-plane coincides with Fourier’s equation∂ 2 u∂x 2 = ∂u∂y(2.3)40

Boundary Value Problems for Differential Equations of Fractional Order 41and in the lower half-plane it coincides with(−y) m ∂2 u∂x 2 = ∂u∂ywhich for y = 0 transforms to an equation of parabolic type.Let Ω be an area bounded by the segments of straight linesand the characteristicsAA 0 : x = 0,A 0 B 0 : y = y 0 ,B 0 B : x = r,(2.4)AC : x − 2m+2(−y) 2 = 0m + 2andBC : x + 2m+2(−y) 2 = rm + 2of the equation (2.4); Ω + = {(x, y) : 0 < x < r, 0 < y < y 0 } be theparabolic part of the mixed area Ω; Ω − be the part of the area Ω, laying inthe lower half-plane y < 0 and bounded by the characteristics AC, 0 ≤ x ≤r2 , BC, r 2≤ x ≤ r and the segment AB; ]0, r[= {(x, 0) : 0 < x < r};{u + (x, y), ∀ (x, y) ∈ Ω + ,u =u − (x, y), ∀ (x, y) ∈ Ω − (2.5),where D l 0x is the operator of fractional integrodifferention of Sturm–Liouvilleof order |l| starting at the point 0.Consider a problem of Tricomi type for the equation (2.1) in Ω, withnonlocal condition of linear conjugation.Problem 1. Find a regular solution of the equation (2.1) in the areasΩ + , Ω − , with the following propertiesu + ∈ C(Ω + ) ∩ C 1( Ω + ∪]0, r[ ) , u − ∈ C(Ω − ) ∩ C 1( Ω − ∪]0, r[ ) , (2.6)u + (x, 0) = u − (x, 0) − λD0x −2 u− (t, 0), 0 ≤ x ≤ r, (2.7)∂(u + − u − )∂y ∣ = 0, 0 < x < r, (2.8)y=0u + (0, y) = ϕ 0 (y), u − (r, y) = ϕ r (y), 0 ≤ y ≤ y 0 , (2.9)u − ∣∣∣AC= ψ(0), 0 < x < r, (2.10)where Ω is the closure of Ω, λ is the spectral parameter, ϕ 0 (y) and ϕ r (y)are given functions of the class C 1 [0, r], and[ x( m + 2) 2/(m+2) ]ψ(x) = u2 , − x4is a given function of the class C 3 [0, r].

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