A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Fig. 1

Fig. 2

Fig. 3

The error norms L2 and L∞ of the time fractional Burgers equation problem using the Crank Nicolson finite difference method with ν = 1_0, Δt = 0_00025 and tf =1 for various values of M

	N = 10	N = 20	N = 40	M = 80
L2 × 103	22.64087326	5.44483424	1.22007333	0.16846258
L∞ × 103	30.49445082	7.70051177	1.72552915	0.23826679

Comparison of results at tf = 1_0 for γ = 0_5, Δt = 0_0001, ν = 1_0 and various mesh sizes

	N = 10	N = 20	N = 40	N = 80
L2 × 103	0.81667090	0.23594859	0.09215464	0.05901177
L∞ × 103	1.13678094	0.32086237	0.12245871	0.09790576

A comparison of the errors for Example 1 at tf =1

	N = 40		N = 80		N = 100
	Present	[9]	Present	[9]	Present	[9]
L2 × 103	1.22007333	1.224329	0.16846258	0.177703	0.04239382	0.052299
L∞ × 103	1.72552915	1.730469	0.23826679	0.253053	0.05996900	0.076541

The error norms L2 and L∞ of Example 1 for N=120, tf = 1_0, Δt = 0_00025 for different values of γ

	γ = 0.1	γ = 0.25	γ = 0.75	γ = 0.9
L2 × 103	0.02411976	0.02490518	0.02669547	0.02579288
L∞ × 103	0.03409905	0.03521115	0.03774592	0.03646791

Comparison of results at tf = 1_0 for ν = 1, Δx = 0_025 and various time steps

	Δt = 0.0001	Δt = 0.00025	Δt = 0.0005	Δt = 0.001
L2 × 103	0.08344454	0.22252258	0.45497994	0.92040447
L∞ × 103	0.23648323	0.59722584	1.19848941	2.40107923

The error norms L2 and L∞ of Example 1 for γ=0_5, Δt=0_00025, tf =1_0, N=40 and various values of ν

	ν = 1	ν = 0.5	ν = 0.1	ν = 0.01	ν = 0.001
L2 × 103	0.41761157	0.51259161	1.03928112	2.13880314	6.46231244
L∞ × 103	0.59040780	0.72342731	1.51280185	4.65707275	22.95302718

Comparison of results at tf = 1_0 for ν=1, N = 40 and various time steps

	Δt = 0.0001	Δt = 0.00025	Δt = 0.0005	Δt = 0.001	Δt = 0.0025	Δt = 0.005
L2 × 103	0.09215464	0.16142590	0.27906906	0.51583267	1.22775834	2.41506007
L∞ × 103	0.12245871	0.24577898	0.47478371	0.93280810	2.30696715	4.59744900