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Theory and Modern Applications

Table 1 Errors and corresponding observation orders of T-ML2 in spatial and temporal directions for Example 1 with different α and λ

From: The implicit midpoint method for Riesz tempered fractional diffusion equation with a nonlinear source term

α

h

Ï„

λ = 0

\(\lambda=\frac {1}{100}\)

λ = 1

∥ε(h,τ)∥

Rate

∥ε(h,τ)∥

Rate

∥ε(h,τ)∥

Rate

1.2

\(\frac{1}{10}\)

\(\frac{1}{10}\)

7.6335e − 04

–

7.5126e − 04

–

1.1638e − 03

–

\(\frac{1}{20}\)

\(\frac{1}{20}\)

1.9911e − 04

1.9388

1.9631e − 04

1.9362

2.8958e − 04

2.0068

\(\frac{1}{40}\)

\(\frac{1}{40}\)

5.1544e − 05

1.9497

5.0882e − 05

1.9479

7.3198e − 05

1.9841

\(\frac{1}{80}\)

\(\frac{1}{80}\)

1.3313e − 05

1.9530

1.3155e − 05

1.9516

1.8860e − 05

1.9565

\(\frac{1}{160}\)

\(\frac{1}{160}\)

3.4300e − 06

1.9565

3.3923e − 06

1.9553

4.8865e − 06

1.9484

1.5

\(\frac{1}{10}\)

\(\frac{1}{10}\)

9.6520e − 04

–

9.6198e − 04

–

2.4219e − 04

–

\(\frac{1}{20}\)

\(\frac{1}{20}\)

2.3429e − 04

2.0425

2.3357e − 04

2.0421

6.6478e − 05

1.8652

\(\frac{1}{40}\)

\(\frac{1}{40}\)

5.6964e − 05

2.0402

5.6791e − 05

2.0401

1.8427e − 05

1.8510

\(\frac{1}{80}\)

\(\frac{1}{80}\)

1.3932e − 05

2.0317

1.3888e − 05

2.0318

5.2173e − 06

1.8205

\(\frac{1}{160}\)

\(\frac{1}{160}\)

3.4330e − 06

2.0208

3.4222e − 06

2.0209

1.4881e − 06

1.8099

1.8

\(\frac{1}{10}\)

\(\frac{1}{10}\)

1.0924e − 03

–

1.0917e − 03

–

8.0464e − 04

–

\(\frac{1}{20}\)

\(\frac{1}{20}\)

2.6152e − 04

2.0625

2.6141e − 04

2.0622

1.9630e − 04

2.0353

\(\frac{1}{40}\)

\(\frac{1}{40}\)

6.2776e − 05

2.0586

6.2754e − 05

2.0585

4.7153e − 05

2.0577

\(\frac{1}{80}\)

\(\frac{1}{80}\)

1.5082e − 05

2.0574

1.5077e − 05

2.0574

1.1234e − 05

2.0695

\(\frac{1}{160}\)

\(\frac{1}{160}\)

3.6282e − 06

2.0555

3.6268e − 06

2.0556

2.6688e − 06

2.0736