Approximation of Semilinear Fractional Cauchy Problem

Abstract

Abstract The semidiscretization methods for solving the Cauchy problem ( 𝐃 t α u ) ( t ) = A u ( t ) + J 1 - α f t , u ( t ) , t ∈ [ 0 , T ] , 0 < α < 1 , u ( 0 ) = u 0 , $(\mathbf {D}_{t}^{\alpha }u)(t) = A u(t) + J^{1-\alpha } f\big (t,u(t)\big ), \quad t \in [0,T], 0 < \alpha <1,\qquad u(0) = u^0,$ with operator A, which generates an analytic and compact resolution family { S α ( t , A ) } t ≥ 0 ${\lbrace S_{\alpha }(t,A)\rbrace _{t\ge 0}}$ , in a Banach space E are presented. It is proved that the compact convergence of resolvents implies the convergence of semidiscrete approximations to an exact solution. We give an analysis of a general approximation scheme, which includes finite differences and projective methods.