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Theory and Modern Applications

Table 2 Numerical errors and convergence orders in spatial direction with \(\tau =\frac{1}{2000}\)

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

 

\(h_{x}=h_{y}\)

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

rate 3

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

rate 4

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

1/4

7.0259e-4

∗

1.4052e-3

∗

1/8

4.3084e-5

4.0274

8.6167e-5

4.0275

1/16

2.6514e-6

4.0223

5.3027e-6

4.0223

1/32

1.3649e-7

4.2798

2.7299e-7

4.2798

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

1/4

7.0294e-4

∗

1.4059e-3

∗

1/8

4.3105e-5

4.0275

8.6210e-5

4.0274

1/16

2.6527e-6

4.0223

5.3054e-6

4.0223

1/32

1.3656e-7

4.2799

2.7312e-7

4.2798

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

1/4

7.0338e-4

∗

1.4067e-3

∗

1/8

4.3132e-5

4.0274

8.6265e-5

4.0274

1/16

2.6547e-6

4.0221

5.3095e-6

4.0221

1/32

1.3705e-7

4.2757

2.7411e-7

4.2757

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

1/4

7.4232e-4

∗

1.4846e-3

∗

1/8

4.5522e-5

4.0274

9.1044e-5

4.0274

1/16

2.8069e-6

4.0195

5.6139e-6

4.0194

1/32

1.5015e-7

4.2245

3.0031e-7

4.2245