Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

07/14/2020
by   Hui Zhang, et al.
0

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 L^2 error estimate in space discretization. The convergence of the fast time-stepping numerical methods is also proved in a simple manner.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/09/2021

Fractional Crank-Nicolson-Galerkin finite element methods for nonlinear time fractional parabolic problems with time delay

A linearized numerical scheme is proposed to solve the nonlinear time fr...
research
01/29/2023

Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models

The discrete gradient structure and the positive definiteness of discret...
research
09/28/2019

Complete monotonicity-preserving numerical methods for time fractional ODEs

The time fractional ODEs are equivalent to the convolutional Volterra in...
research
10/10/2022

A smoothing analysis for multigrid methods applied to tempered fractional problems

We consider the numerical solution of time-dependent space tempered frac...
research
12/16/2022

Roundoff error problem in L2-type methods for time-fractional problems

Roundoff error problems have occurred frequently in interpolation method...
research
01/12/2021

Sharp pointwise-in-time error estimate of L1 scheme for nonlinear subdiffusion equations

An essential feature of the subdiffusion equations with the α-order time...
research
04/15/2020

Regularization of backward time-fractional parabolic equations by Sobolev equations methods

It is well-known that backward parabolic equations in which the initial ...

Please sign up or login with your details

Forgot password? Click here to reset