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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