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Theory and Modern Applications

Table 10 Errors and corresponding observation orders at \(t=1\), \(\gamma _{3}=0\)

From: Second-order numerical methods for the tempered fractional diffusion equations

α

h = τ

λ = 1/50

λ = 0

λ = 2

\(\|e\|_{L_{2}}\)

Order

\(\|e\|_{L_{2}}\)

Order

\(\|e\|_{L_{2}}\)

Order

1.2

1/10

3.6036e−04

 

3.9643e−04

 

3.8557e−04

 

1/20

9.3836e−05

1.9412

9.8756e−05

2.0051

1.0825e−04

1.8326

1/40

2.4100e−05

1.9611

2.4944e−05

1.9852

2.7754e−05

1.9636

1/80

6.1263e−06

1.9759

6.2947e−06

1.9865

6.9556e−06

1.9964

1.5

1/10

4.8058e−04

 

4.9188e−04

 

2.3030e−04

 

1/20

1.2712e−04

1.9186

1.2939e−04

1.9266

6.1371e−05

1.9079

1/40

3.2771e−05

1.9557

3.3285e−05

1.9588

1.5816e−05

1.9562

1/80

8.3331e−06

1.9755

8.4560e−06

1.9768

4.0057e−06

1.9813

1.8

1/10

5.0678e−04

 

5.1402e−04

 

1.7371e−04

 

1/20

1.3156e−04

1.9456

1.3326e−04

1.9476

4.5103e−05

1.9454

1/40

3.3388e−05

1.9783

3.3800e−05

1.9791

1.1761e−05

1.9392

1/80

8.4106e−06

1.9890

8.5114e−06

1.9896

3.0136e−06

1.9645