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Theory and Modern Applications

Table 1 The maximum errors at time \(\pmb{t=1}\) and convergence orders in spatial and temporal directions with \(\pmb{h=\tau=\frac{1}{N+1}}\) for Example  1 with different α , and \(\pmb{\lambda=1.0}\) , the parameters \(\pmb{\gamma_{1}=0.7, 0.75, 0.80}\) , respectively, and \(\pmb{\gamma_{2}}\) and \(\pmb{\gamma _{3}}\) are selected in the set \(\pmb{S_{1}^{\alpha}(\gamma_{1},\gamma _{2},\gamma_{3})}\)

From: Stability and convergence of the Crank-Nicolson scheme for a class of variable-coefficient tempered fractional diffusion equations

α

N  + 1

\(\boldsymbol{\gamma_{1}=0.70}\)

\(\boldsymbol{\gamma_{1}=0.75}\)

\(\boldsymbol{\gamma_{1}=0.80}\)

e ( h , Ï„ )

Order

e ( h , Ï„ )

Order

e ( h , Ï„ )

Order

1.2

26

1.6718e−06

-

2.0479e−06

-

2.4267e−06

-

27

3.9099e−07

2.10

4.7155e−07

2.12

5.5211e−07

2.14

28

9.4311e−08

2.05

1.1268e−07

2.07

1.3106e−07

2.07

29

2.3150e−08

2.03

2.7526e−08

2.03

3.1903e−08

2.04

210

5.7343e−09

2.01

6.8018e−09

2.02

7.8694e−09

2.02

1.5

26

6.3777e−07

-

8.4512e−07

-

1.0529e−06

-

27

1.5554e−07

2.04

2.1700e−07

1.96

2.7849e−07

1.92

28

3.8376e−08

2.02

5.4938e−08

1.98

7.1505e−08

1.96

29

9.5342e−09

2.01

1.3826e−08

1.99

1.8118e−08

1.98

210

2.3761e−09

2.00

3.4680e−09

2.00

4.5598e−09

1.99

1.8

26

2.8859e−07

-

9.0103e−08

-

2.0052e−07

-

27

9.1854e−08

1.65

3.2151e−08

1.49

3.9194e−08

2.36

28

2.5751e−08

1.83

9.3079e−09

1.79

8.5674e−09

2.19

29

6.7699e−09

1.93

2.4899e−09

1.90

1.9963e−09

2.10

210

1.7328e−09

1.97

6.4317e−10

1.95

4.8152e−10

2.05