Table 1 Comparison between the exact and approximation solutions for \(\psi (\chi ,\sigma ,t)\)

0

0

0

0

0

0.8000

0.8777

0.0777

0.8148

0.0148

1.6000

1.7554

0.1554

1.6295

0.0295

2.4000

2.6331

0.2331

2.4443

0.0443

3.2000

3.5108

0.3108

3.2591

0.0591

4.0000

4.3886

0.3886

4.0739

0.0739

4.8000

5.2663

0.4663

4.8886

0.0886

5.6000

6.1440

0.5440

5.7034

0.1034

6.4000

7.0217

0.6217

6.5182

0.1182

7.2000

7.8994

0.6994

7.3330

0.1330

8.0000

8.7771

0.7771

8.1477

0.1477