Approximation of Semilinear Fractional Cauchy Problem

Ru Liu; Miao Li; Sergey Piskarev

doi:10.1515/cmam-2015-0001

Published by De Gruyter February 3, 2015

Approximation of Semilinear Fractional Cauchy Problem

Ru Liu , Miao Li and Sergey Piskarev

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2015-0001

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Abstract

The semidiscretization methods for solving the Cauchy problem

(𝐃tαu)(t)=Au(t)+J1-αft,u(t),t∈[0,T],0<α<1,u(0)=u0,

with operator A, which generates an analytic and compact resolution family {Sα(t,A)}t≥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.