Approximation of semilinear fractional Cauchy problem: II
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Approximation of semilinear fractional Cauchy problem: II Ru Liu1 · Sergey Piskarev2 Received: 18 August 2018 / Accepted: 17 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider the semidiscrete approximation of the Cauchy problem ( ) (𝐃𝛼t u)(t) = Au(t) + f t, u(t) , t ∈ (0, T], u(0) = u0 , 0 < 𝛼 < 1, on a Banach space, where the operator A generates an analytic and compact resolvent family {S𝛼 (t, A)}t≥0 and the function f (⋅, ⋅) is strongly continuous. We give an analysis of a general approximation scheme, which includes finite differences and projective methods. Keywords Fractional Cauchy problem · Semilinear equation · 𝛼-Times resolvent family · Analytic 𝛼-times · Compact 𝛼-times resolvent family · (𝛼, 𝛼)-Times resolvent family
1 Introduction and preliminaries In [19] we discussed the semidiscrete approximation of the Cauchy problem
Communicated by Abdelaziz Rhandi. 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. * Ru Liu [email protected] Sergey Piskarev [email protected] 1
College of Information Science and Engineering, Chengdu University, Chengdu 610106, Sichuan, People’s Republic of China
2
Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow, Russia 119991
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R. Liu, S. Piskarev
(𝐃𝛼t u)(t) = Au(t) + J 1−𝛼 f (t, u(t)), 0 < t ≤ T; u(0) = u0 ,
(1)
on a Banach space E, where 𝐃𝛼t is the Caputo–Dzhrbashyan derivative. The operator A generates an analytic and compact resolvent family S𝛼 (t, A) and the function f (⋅, ⋅) is smooth enough in both arguments. In this paper, we consider the well-posed Cauchy problem
(𝐃𝛼t u)(t) = Au(t) + f (t, u(t)), 0 < t ≤ T; u(0) = u0 ,
(2)
on a Banach space E. The operator A and the function f (⋅, ⋅) satisfy the same conditions as above, respectively. Remark 1 It might seem that we consider an equation which is just a little bit different from (1) in [19]. However, the approximation of (2) is more complicated than that of (1). The mild solution of (1) can be represented by t
S𝛼 (t, A)u0 +
S𝛼 (t − s, A)f (s, u(s))ds,
∫0
while the mild solution of (2) is t
S𝛼 (t, A)u0 +
∫0
P𝛼 (t − s, A)f (s, u(s))ds.
S𝛼 (t, A) is strongly continuous for t ≥ 0 . However, the singularity of P𝛼 (t, A) at 0 brings us main difficulty. Let us recall some definitions (see [12]). The fractional integral of order 𝛼 > 0 is defined by
(J 𝛼 q)(t) ∶= (g𝛼 ∗ q)(t), {
t > 0,
t𝛼−1 , Γ(𝛼)
t > 0, and Γ(𝛼) is the Gamma function. The Riemann–Liou0, t ≤ 0, ville derivative of order 𝛼 > 0 is ( )m ( 𝛼 ) d (J m−𝛼 q)(t), Dt q (t) = dt
where g𝛼 (t) ∶=
and the Caputo–Dzhrbashyan fractional derivative of order 𝛼 > 0 is defined by
(𝐃𝛼t q)(t)
m−1 ∑ ( 𝛼 ) = Dt q (t) − k=0
13
q(k) (0) tk−𝛼 . Γ(k − 𝛼 + 1)
Approximation of semilinear fractional Cauchy problem: II
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