# Table 2 The numerical results for Example 2 with $${x}=0.5$$, $${M} =2$$, $${N} =3$$

r Exact solution Presented method Absolute error Method [47]
HBPBP BP
$$\underline{{u}} ( {x}, {r} )$$
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
(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