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Approximation of semilinear fractional Cauchy problem: II

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Abstract

We consider the semidiscrete approximation of the Cauchy problem

$$\begin{aligned} ({\mathbf {D}}_{t}^{\alpha }u)(t) = A u(t) +f\big (t,u(t)\big ), \quad t \in (0,T],\ u(0) = u^0,\ 0< \alpha <1, \end{aligned}$$

on a Banach space, where the operator A generates an analytic and compact resolvent family \(\{S_{\alpha }(t,A)\}_{t\ge 0}\) and the function \(f(\cdot , \cdot )\) is strongly continuous. We give an analysis of a general approximation scheme, which includes finite differences and projective methods.

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Authors and Affiliations

  1. College of Information Science and Engineering, Chengdu University, Chengdu, 610106, Sichuan, People’s Republic of China

    Ru Liu

  2. Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow, Russia, 119991

    Sergey Piskarev

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  1. Ru Liu
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  2. Sergey Piskarev
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Correspondence to Ru Liu.

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Communicated by Abdelaziz Rhandi.

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Ru Liu is supported by Tianyuan Youth Fund of Mathematics, NSFC (No. 11626046) and Scientific Research Starting Foundation (Chengdu University, No. 2081915055), Sergey Piskarev is supported by grants of Russian Foundation for Basic Research 17-51-53008 and 16-01-00039-a.

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Liu, R., Piskarev, S. Approximation of semilinear fractional Cauchy problem: II. Semigroup Forum 101, 751–768 (2020). https://doi.org/10.1007/s00233-020-10136-z

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