FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

CHAPTER 3Solving Two-Point Boundary Value Problemsfor Fractional Differential Equations (FBVPs)1. The Basic ConceptsIn [8] and [33], the existence and uniqueness of solutions, and numericalmethods for FBVPs with Caputo’s derivatives and with Riemann–Liouvillederivatives are studied, respectively.FBVPs with Caputo’s derivatives:andCa D γ t y(t) + f (t, y(t)) = 0, a < t < b, 1 < γ ≤ 2, (3.1)y(a) = α, y(b) = β (3.2)Ca D γ t y(t) + g ( t, y(t), C a D θ t y(t) ) = 0, a < t < b, (3.3)y(a) = α, y(b) = β, 1 < γ ≤ 2, 0 < θ ≤ 1, (3.4)where y : [a, b] ↦→ R, f : [a, b] × R ↦→ R, g : [a, b] × R × R ↦→ R are continuousand satisfy Lipschitz conditions|f(t, x) − f(t, y)| ≤ K f |x − y|, (3.5)∣∣g(t, x 2 , y 2 ) − g(t, x 1 , y 1 ) ∣ ≤ K g |x 2 − x 1 | + L g |y 2 − y 1 | (3.6)with Lipschitz constants K f , K g , L g > 0.FBVPs with Riemann–Liouville derivatives:andRLa D γ t y(t) + f (t, y(t)) = 0, a < t < b, 1 < γ ≤ 2, (3.7)y(a) = 0, y(b) = β, (3.8)RLa D γ t y(t) + g ( t, y(t), RLa Dt θ y(t) ) = 0, a < t < b, (3.9)y(a) = 0, y(b) = β, 1 < γ ≤ 2, 0 < θ ≤ γ − 1, (3.10)where y : [a, b] ↦→ R, f : [a, b] × R ↦→ R, g : [a, b] × R × R ↦→ R are continuousand satisfy Lipschitz conditions (3.5) and (3.6), respectively.In order to state the problems in concern, we introduce the definitionsand properties of Caputo’s derivatives and Riemann–Liouville fractionalintegrals in [8].Definition 3.1 (see [29]). Let γ > 0, n−1 < γ ≤ n and let C n [a, b] :={y(t) : [a, b] → R; y(t) has a continuous n-th derivative}.66