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Table 1 The numerical results for Example 1 with \({x}=0.5\), \({M} =2\), \({N} =3\)

From: Study of hybrid orthonormal functions method for solving second kind fuzzy Fredholm integral equations

r Exact solution HBT method [46] Block-pulse method [45] Presented method Absolute error
  \(\underline{u} ( x,r )\)     
0 0.000000000 0.000000000 0.007956 0.000000000 8.58708660e−11
0.1 0.050000000 0.050000000 0.056347 0.050000000 6.79634155e−11
0.2 0.100000000 0.100000000 0.104737 0.100000000 5.21343024e−11
0.3 0.150000000 0.150000000 0.153128 0.150000000 3.83835130e−11
0.4 0.200000000 0.200000000 0.201519 0.200000000 2.67110889e−11
0.5 0.250000000 0.250000000 0.266040 0.250000000 1.53477231e−12
0.6 0.300000000 0.300000000 0.314430 0.300000000 8.80506779e−13
0.7 0.350000000 0.350000000 0.362820 0.350000000 4.41124914e−13
0.8 0.400000000 0.400000000 0.411210 0.400000000 2.16571205e−13
0.9 0.45000000 0.450000000 0.359603 0.450000000 1.03909548e−11
  (x,r)     
0 1.000000000 1.000000000 1.024160 1.000000000 1.00612851e−11
0.1 0.950000000 0.950000000 0.975770 0.950000000 9.72248948e−12
0.2 0.900000000 0.900000000 0.927379 0.900000000 7.29647454e−12
0.3 0.850000000 0.850000000 0.878988 0.850000000 2.78301826e−12
0.4 0.800000000 0.800000000 0.830598 0.800000000 3.81765730e−12
0.5 0.750000000 0.750000000 0.766077 0.750000000 0.00000000e+00
0.6 0.700000000 0.700000000 0.717986 0.700000000 3.09223758e−12
0.7 0.650000000 0.650000000 0.669290 0.650000000 6.18458618e−12
0.8 0.600000000 0.600000000 0.630905 0.600000000 9.27682375e−12
0.9 0.550000000 0.550000000 0.572514 0.550000000 4.72755612e−11