Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 05, pp. 1-13.

Title: theta-scheme for solving Caputo fractional differential equations

Authors: Thai Son Doan (Vietnam Academy of Science and Tech., Ha Noi, Vietnam)
         Phan Thi Huong (Le Quy Don Technical Univ., Ha Noi, Vietnam)
         Peter E. Kloeden (Univ. Tubingen, Germany)
  
Abstract:
We formulate a $\theta$-numerical scheme for solving
Caputo fractional differential equations (Caputo FDEs) of order 
$\alpha\in(0,1)$, with vector fields satisfying  a standard 
Lipschitz continuity condition in the state variable and a 
H\"older continuity condition in the time variable. 
The  convergence rate is  established and a numerical example 
is given to  illustrate the theoretical results.  
The scheme obtained includes the explicit ($\theta=0$) 
and implicit ($\theta=1$) counterparts of Euler-like schemes 
for Caputo FDEs  known in the literature as  the Adams-Bashford and 
Adams-Moulton schemes, respectively, and essentially linearly 
interpolates them.


Submitted August 09, 2024. Published January 09, 2025.
Math Subject Classifications: 34A05, 65L99, 65R20.
Key Words: Caputo fractional differential equations; theta-scheme; Euler scheme; 
           Adams-Bashford scheme; Adams-Moulton scheme.