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Theory and Modern Applications

Table 2 Exact solution and absolute errors for Example 1 at time \(t=1\), \(\beta=1\), fixed \(h=0.015625\) and \(\tau=0.01\) using FDSF and FVSF

From: A new formulation of finite difference and finite volume methods for solving a space fractional convection–diffusion model with fewer error estimates

x

Exact solution

Absolute error

Absolute error

 

FDSF

FVSF

1.

0.

7.193791 × 10−5

5.814751 × 10−6

1.015625

−1.312701 × 10−10

1.438076 × 10−5

1.162322 × 10−7

1.03125

−2.622240 × 10−10

2.1553847 × 10−5

1.741653 × 10−7

1.046875

−3.925462 × 10−10

2.8706087 × 10−5

2.318803 × 10−7

1.0625

−5.219227 × 10−10

3.5830568 × 10−5

2.893150 × 10−7

1.078125

−6.500419 × 10−10

4.2920399 × 10−5

3.464091 × 10−7

1.09375

−7.765951 × 10−10

4.9968719 × 10−5

4.031040 × 10−7

1.109375

−9.012773 × 10−10

5.696871 × 10−5

4.593420 × 10−7

1.125

−1.023788 × 10−9

6.391360 × 10−5

5.150668 × 10−7

1.140625

−1.143833 × 10−9

7.079666 × 10−5

5.702229 × 10−7

1.15625

−1.261122 × 10−9

7.761125 × 10−5

6.247557 × 10−7

1.171875

−1.375372 × 10−9

8.435077 × 10−5

6.786116 × 10−7

1.1875

−1.486310 × 10−9

9.100869 × 10−5

7.317379 × 10−7

1.203125

−1.593667 × 10−9

9.757858 × 10−5

7.840830 × 10−7

1.21875

−1.697184 × 10−9

10.40540 × 10−5

8.355962 × 10−7

1.234375

−1.796613 × 10−9

11.04289 × 10−5

8.862276 × 10−7

1.25

−1.891714 × 10−9

11.66969 × 10−5

9.359287 × 10−7