FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

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