FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

74 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangDue to (3.44)–(3.45), we consider {|y(t n ) − y n |} N n=0. Then we getmax |y(t n) − y n | = O(h r ). (3.47)0≤n≤NAs to the linear case of (3.7)–(3.8), we only need to replace (3.42) and(3.43) with the initial value conditionsRLa D γ−1t y(t) ∣ = ξ 1 , lim J 2−γt=a t→a + a y(t) = 0, (3.48)D γ−1y(t) ∣ = ξ 2 , ;t=aRLatlim J 2−γt→a + a y(t) = 0. (3.49)Again, the shooting procedures for linear FBVPs (3.3)–(3.4) and (3.9)–(3.10) are similar to that of (3.1)–(3.2) and (3.7)–(3.8), respectively.3.2. Shooting method for nonlinear problems. For nonlinearFBVP (3.1) and (3.7) with homogeneous boundary value conditions (3.41),we consider the following FIVP:Ca D γ t y(t) + f(t, y(t)) = 0, a ≤ t ≤ b, 1 < γ ≤ 2, (3.50)y(a) = 0, y ′ (a) = ξ, (3.51)The FIVPs (3.50) is corresponding to FBVP (3.1), (3.41) with analyticsolutions denoted as y(t; ξ). In general, y(b; ξ) ≠ 0. Once a zero point ξ ∗of φ(ξ) := y(b; ξ) is found, one obtains y(t; ξ ∗ ), the solution of FBVP (3.1),(3.41). When y(t; ξ), and hence φ(ξ) are continuously differentiable withrespect to ξ, Newton’s method can be employed to determine ξ ∗ . Startingwith an initial approximation ξ (0) , one gets ξ (k) as follows:ξ (k+1) = ξ (k) − φ(ξ(k) )φ ′ (ξ (k) ) . (3.52)On one hand, y(b; ξ), hence φ(ξ) can be determined by solving FIVP (3.50)numerically. On the other hand, it is easy to check that w(t; ξ) := ∂y(t;ξ)∂ξisa solution of the following FIVP (refer to [30]):Ca D γ ∂f(t, y(t; ξ))t w(t; ξ) + w(t; ξ) = 0, (3.53)∂yw(a; ξ) = 0, w ′ (a; ξ) = 1, a ≤ t ≤ b, 1 < γ ≤ 2. (3.54)So we can compute w(b; ξ), i.e., φ ′ (ξ) by solving FIVPs (3.53) numerically.However, considering the computation complexity for ∂f(t,y(t;ξ))∂yin practice,we usually calculate the difference quotient△φ(ξ (k) ) := φ(ξ(k) + △ξ (k) ) − φ(ξ (k) )△ξ (k)instead of the derivative φ ′ (ξ (k) ) itself, where △ξ (k) is a sufficiently smallnumber. And the formula (3.52) is replaced byξ (k+1) = ξ (k) − φ(ξ(k) )△φ(ξ (k) ) . (3.55)