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Theory and Modern Applications

Table 1 Numerical errors and convergence orders in time direction with \(h=\frac{1}{128}\)

From: An efficient numerical algorithm for solving the two-dimensional fractional cable equation

 

Ï„

\(e_{2}(h,\tau)\)

rate 1

\(e_{\infty }(h,\tau)\)

rate 2

\(\gamma _{1}=0.9\), \(\gamma _{2}=0.8\)

1/20

3.1063e-4

∗

6.2126e-4

∗

1/40

7.7663e-5

1.9999

1.5532e-4

1.9998

1/80

1.9416e-5

2.0000

3.8832e-5

1.9999

1/160

4.8537e-6

2.0001

9.7074e-6

2.0001

\(\gamma _{1}=0.8\), \(\gamma _{2}=0.6\)

1/20

3.0836e-4

∗

6.1672e-4

∗

1/40

7.7136e-5

1.9991

1.5427e-4

1.9991

1/80

1.9291e-5

1.9995

3.8582e-5

1.9995

1/160

4.8233e-6

1.9998

9.6467e-6

1.9998

\(\gamma _{1}=0.6\), \(\gamma _{2}=0.2\)

1/20

2.9879e-4

∗

5.9758e-4

∗

1/40

7.5111e-5

1.9920

1.5022e-4

1.9920

1/80

1.8851e-5

1.9944

3.7702e-5

1.9944

1/160

4.7253e-6

1.9961

9.4507e-6

1.9965

\(\gamma _{1}=0.2\), \(\gamma _{2}=0.6\)

1/20

2.1680e-4

∗

4.3360e-4

∗

1/40

5.6905e-5

1.9297

1.1381e-4

1.9297

1/80

1.4814e-5

1.9415

2.9629e-5

1.9415

1/160

3.8312e-6

1.9511

7.6624e-6

1.9511