FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

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)