2020/07/14 by Hui Zhang, Fanhai Zeng, Zhang, Hui +5
Mathematics · #35E15 #35K55 #65M12 #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2007.07015
openalex publication_date 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1986, Dixon and McKee developed a discrete fractional Grönwall inequality [Z. Angew. Math. Mech., 66 (1986), pp. 535--544], which can be seen as a generalization of the classical discrete Grönwall inequality. However, this generalized discrete Grönwall inequality has not been widely applied in the numerical analysis of the time-stepping methods for the time-fractional evolution equations. The main purpose of this paper is to show how to apply the generalized discrete Grönwall inequality to prove the convergence of a class of time-stepping numerical methods for time-fractional nonlinear subdiffusion equations, including the popular fractional backward difference type methods of order one and two, and the second-order fractional Crank-Nicolson type methods. We obtain the optimal L2 error estimate in space discretization. The convergence of the fast time-stepping numerical methods is also proved in a simple manner.