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Table 3 Numerical results of Example 4.1 using the framelet systems \({{X(\Omega _{1})}}\) and \({{X(\Omega _{2})}}\) for \(m=3\) and \(\lambda =3.75\)

From: Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications

x Exact \({{\mathcal{W}_{m}u}}\) via \({{X(\Omega _{1})}}\) \({{\mathcal{W}_{m}u}}\) via \({{X(\Omega _{2})}}\) \({{\mathcal{M}u}}\), \(a=3=p^{*}=3\)
0.0 1.00000 1.00000 1.00000 1.00000
0.1 1.10527 1.10521 1.10514 1.10518
0.2 1.22140 1.22152 1.22146 1.22159
0.3 1.34986 1.35025 1.35022 1.35027
0.4 1.49182 1.49282 1.49280 1.49354
0.5 1.64872 1.65086 1.65077 1.65217
0.6 1.82212 1.82613 1.82599 1.82669
0.7 2.01375 2.02059 2.02044 2.02240
0.8 2.22554 2.23642 2.23628 2.23719
0.9 2.45960 2.47597 2.47582 2.47665
1.0 2.71828 2.72536 2.71663 2.73001