Table 2 Computed maximum absolute errors for Example 1 by the proposed method on piecewise mesh

\(\alpha = \frac{1}{12},\,\,\beta = \frac{5}{12}\)

\(10^{ - \,3}\)

2.9485e−04

2.0460e−05

1.3407e−06

8.5692e−08

5.4143e−09

3.4022e−10

\(10^{ - \,4}\)

3.2535e−04

4.1312e−05

4.6331e−06

4.7689e−07

4.6128e−08

4.2567e−09

\(10^{ - \,6}\)

3.1203e−04

4.7604e−05

4.6331e−06

1.2766e−06

1.3469e−07

1.0771e−08

\(10^{ - \,8}\)

3.1070e−04

6.2649e−05

8.7563e−06

3.4607e−06

7.6676e−07

1.5275e−07

\(10^{ - \,10}\)

3.1057e−04

6.4374e−05

1.4857e−05

3.8392e−06

9.4147e−07

2.2909e−07

\(10^{ - \,12}\)

3.1055e−04

6.4549e−05

1.5662e−05

3.8792e−06

9.6101e−07

2.3864e−07

\(10^{ - \,16}\)

3.1055e−04

6.4568e−05

1.5754e−05

3.8836e−06

9.6318e−07

2.3971e−07

\(\alpha = \frac{1}{3},\,\,\beta = \frac{1}{6}\)

\(10^{ - \,3}\)

1.6639e−02

4.5091e−03

1.1885e−03

3.0595e−04

7.7666e−05

1.9568e−05

\(10^{ - \,4}\)

1.7402e−02

6.3386e−03

2.1634e−03

7.0684e−04

2.2270e−04

6.8177e−05

\(10^{ - \,6}\)

1.6980e−02

6.1878e−03

2.1102e−03

6.8962e−04

2.1718e−04

6.6467e−05

\(10^{ - \,8}\)

1.6941e−02

6.1728e−03

2.1049e−03

6.8790e−04

2.1663e−04

6.6296e−05

\(10^{ - \,10}\)

1.6936e−02

6.1713e−03

2.1043e−03

6.8772e−04

2.1658e−04

6.6279e−05

\(10^{ - \,12}\)

1.6936e−02

6.1711e−03

2.1043e−03

6.8771e−04

2.1657e−04

6.6277e−05

\(10^{ - \,16}\)

1.6936e−02

6.1711e−03

2.1043e−03

6.8771e−04

2.1657e−04

6.6277e−05