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Theory and Modern Applications

Table 2 Errors and corresponding observation orders at \(t=1\), \(\alpha =1.5\)

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

λ

h = τ

\(\gamma _{3}=-0.142\)

\(\gamma _{3}=0\)

\(\gamma _{3}=0.333\)

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

Order

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

Order

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

Order

1/50

1/10

7.7227e−03

 

1.0465e−02

 

5.3286e−02

 

1/20

2.0245e−03

1.9315

2.6046e−03

2.0064

1.3526e−02

1.9780

1/40

5.1203e−04

1.9833

6.4995e−04

2.0027

3.3823e−03

1.9997

1/80

1.2836e−04

1.9960

1.6239e−04

2.0009

8.4480e−04

2.0013

0

1/10

5.7263e−03

 

1.1908e−02

 

5.3855e−02

 

1/20

1.5271e−03

1.9068

2.9592e−03

2.0087

1.3568e−02

1.9889

1/40

3.8755e−04

1.9783

7.3811e−04

2.0033

3.3854e−03

2.0028

1/80

9.7221e−05

1.9950

1.8439e−04

2.0011

8.4508e−04

2.0022

2

1/10

2.5444e−03

 

6.6079e−03

 

2.2144e−02

 

1/20

7.8955e−04

1.6882

1.7420e−03

1.9234

6.9946e−03

1.6626

1/40

2.0774e−04

1.9263

4.4145e−04

1.9804

1.8798e−03

1.8956

1/80

5.2509e−05

1.9841

1.1072e−04

1.9953

4.7865e−04

1.9735