Table 2 The numerical results for Example 2 with \({x}=0.5\), \({M} =2\), \({N} =3\)

\(\underline{{u}} ( {x}, {r} )\)

0

0.000000

0.000000

8.587086597e−11

0.001190

0.104322

0.1

0.050000

0.050000

6.796341545e−11

0.002138

0.024766

0.2

0.100000

0.100000

5.213430243e−11

0.004753

0.025467

0.3

0.150000

0.150000

3.838351303e−11

0.005232

0.030005

0.4

0.200000

0.200000

2.671108890e−11

0.004012

0.085691

0.5

0.250000

0.250000

1.534772309e−12

0.000891

0.070301

0.6

0.300000

0.300000

8.805067786e−13

0.002630

0.084357

0.7

0.350000

0.350000

4.411249144e−13

0.003999

0.099502

0.8

0.400000

0.400000

2.165712054e−13

0.000248

0.054350

0.9

0.450000

0.450000

1.039095476e−11

0.003274

0.066229

1.0

0.500000

0.500000

1.773514668e−11

0.002265

0.012674

u̅(x,r)

0

1.000000

1.000000

1.006128514e−11

0.002730

0.095432

0.1

0.950000

0.950000

9.722489480e−12

0.001276

0.083485

0.2

0.900000

0.900000

7.296474536e−12

0.001000

0.072324

0.3

0.850000

0.850000

2.783018260e−12

0.008388

0.062431

0.4

0.800000

0.800000

3.817657301e−12

0.001192

0.059004

0.5

0.750000

0.750000

0.000000000e+00

0.002449

0.055920

0.6

0.700000

0.700000

3.092237577e−12

0.008295

0.061354

0.7

0.650000

0.650000

6.184586177e−12

0.002286

0.075632

0.8

0.600000

0.600000

9.276823754e−12

0.000710

0.059959

0.9

0.550000

0.550000

4.727556124e−11

0.005462

0.040971

1.0

0.500000

0.500000

5.854872143e−11

0.001111

0.028821