FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

28 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangwhereM ε u =K ε (x, t) =N ε =˜K ε (x, t) =∫ x0K ε (x, t)u(t) dt, (1.6){(x − t) 1+ε , t < x,0, t ≥ x∫ 10˜K ε (x; t)u(t) dt,{x 1+ε (1 − t) 1+ε , t ≠ 1,0, t = 1.(1.7)(1.8)Considering the disturbance of the operator A ε , we write(A − A ε )u = (M − M ε )u − (N − N ε )u.First, we deal with (M − M ε )u. Clearly,(M − M ε )u =∫ 10[K(x, t) − Kε (x, t) ] u(t) dt.Since{(x − t)[1 − (x − t) ε ], t < xK(x, t) − K ε (x, t) =0, t ≥ x =⎧⎨[(x−t) ε ln(x−t) +ε=2 ln2 (x−t)+ · · · +ε n lnn (x−t)]+ · · · , t < x,⎩1!2!n!0, t ≥ x,we havewhereThusM ε u =(M − M ε )u = ε∫ 10∫ 10∫1K 1 (x, t)u(t) dt + · · · + ε n⎧⎨(x − t) ln n (x − t), t < x,K n (x, t) = n!⎩0, t ≥ x.K(x, t)u(t) dt−ε∫ 100K n (x, t)u(t) dt + · · · ,∫1K 1 (x, t)u(t) dt−· · ·−ε n K n (x, t)u(t) dt−· · · .0