FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Since∣∣f(t, u 2 ) − f(t, u 1 ) ∣ ∣ = ∣ ∣ sin(u 2 ) − sin(u 1 ) ∣ ∣ ≤ |u 2 − u 1 |,78 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa Tang2(b − a) γ [K gΓ(γ + 1) + L g(b − a) γ−θΓ(2 − θ)Γ(γ + 1)(b −]a)γ−θ+ ≈ 0.96 < 1,Γ(γ − θ + 1)there exists a unique solution for this FBVP according to Theorem 3.1. Infact, one can easily check that y(t) = t(1 − t) is the analytical solution. Theerrors between the numerical solution (obtained by using shooting methodmentioned above) and the analytical solution at mesh points are plotted inFigure 1. The true solution (denoted by real line) and numerical solution(denoted by □) on equispaced mesh are plotted in Figure 2.whereExample 3.2. Consider the following nonlinear FBVPRL−1 Dt 1.5 y(t) + sin(y) + r(t) = 0, −1 < t < 1,y(−1) = 0, y(1) = 0,r(t) = sin(C t+1 (2, 2π)) + C t+1 (0.5, 2π),C t−a (α, ω) = (t − a) α∞∑j=0(−1) j (ω(t − a)) 2jΓ(α + 2j + 1).3 x 10−32.521.5error10.50−0.5−1−1.5−2−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1tFigure 3. the shooting error for Example 3.2K f = 1, a = −1, b = 1, γ = 1.5,K f(γ − 1) γ−1 (b − a) γγ γ Γ(γ + 1)≈ 0.82 < 1,