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

Advances in Continuous and Discrete Models

Theory and Modern Applications

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

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
	u̅(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