FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

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72 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa Tang3. Shooting Methods for FBVPsIn this section, single shooting methods are applied to solve the FBVPs(3.1)–(3.2), (3.3)–(3.4), (3.7)–(3.8) and (3.9)–(3.10) numerically.According to the idea of the shooting method, a FBVP is turned into afractional initial value problem (FIVP) which can be solved by some suitablenumerical method (see [26]–[28], [31]). We write down the correspondingprocedure for FBVPs.Denote the corresponding initial value conditions of FBVPs (3.1)–(3.2)and (3.3)–(3.4) asy(a) = a 0 , y ′ (a) = a 1 , a 0 , a 1 ∈ R. (3.31)Then FBVPs (3.1)–(3.2) and (3.3)–(3.4) can turn into FIVPs (3.1), (3.31)and (3.3), (3.31), respectively.Usually, not all a k (k = 0, 1) are equal to zero, in other words, the initialvalue conditions (3.31) are inhomogeneous. Settingz(t) = y(t) − a 0 − a 1 (t − a), (3.32)(3.1), (3.31) and (3.3)–(3.31) can be transformed into another FIVPs withhomogeneous initial value conditions.For a given equispaced mesh a = t 0 < t 1 < · · · < t N = b with stepsizeh = (b − a)/N, we give a fractional backward difference scheme of order one(refer to [31]) to solve FIVPs (3.1) and (3.3) with homogeneous initial valueconditions y(a) = 0, y ′ (a) = 0:m∑z m = −h γ f(t m , z m ) − ω k z m−k , (3.33)andi=0k=1z m = −h γ g(t m , z m , 1 ∑m ) m∑h θ ˜ω i z m−i − ω k z m−k , (3.34)where(ω 0 = 1, ω k = 1 − γ + 1 )ω k−1 , k = 1, 2, . . . , N, (3.35)k(˜ω 0 = 1, ˜ω i = 1 − θ + 1 )˜ω i−1 , i = 1, 2, . . . , N. (3.36)iThe case for (3.7)–(3.8) and (3.9)–(3.10) is similar. The reader shouldnote that the initial values take the following formRLa D γ−1t y(a) = b 1 ∈ R, lim t→a + a y(t) = b 2 ∈ R. (3.37)It is easy to check thatUsually, b 1 ≠ 0 and letk=1b 2 = limt→a + J 2−γa y(t) = 0. (3.38)z(t) = y(t) − b 1(t − a) γ−1Γ(γ)(3.39)
Boundary Value Problems for Differential Equations of Fractional Order 73in (3.7), (3.37) and (3.9), (3.37). Then they are transformed into FIVPswith homogeneous initial conditions. (3.33) or (3.34) is also employed tosimulate them.3.1. Shooting method for linear problems. The linear case of fractionaltwo-point boundary value problem (3.1) with homogeneous boundaryvalue conditionsCa D γ t y(t) + c(t)y(t) + d(t) = 0, a < t < b, 1 < γ ≤ 2, (3.40)y(a) = 0, y(b) = 0 (3.41)where c(t), d(t) ∈ C 0 [a, b].According to Theorem 3.1, ifb − a
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