Table 5.

Numerical results of Example 5 using the framelet systems $\mathbf{X}\left({\Xi }_{\mathbf{1}}\right)$ and $\mathbf{X}\left({\Xi }_{\mathbf{2}}\right)$ for $m=4$.

x	Exact	${\mathcal{E}}_{\mathit{m}}\mathit{u}$ via $\mathbf{X}\left({\mathit{\Xi }}_1\right)$	${\mathcal{E}}_{\mathit{m}}\mathit{u}$ via $\mathbf{X}\left({\mathit{\Xi }}_2\right)$
0.1	0.01	1.62034 × ${10}^{-8}$	8.34395 × ${10}^{-9}$
0.2	0.04	1.03814 × ${10}^{-7}$	5.23015 × ${10}^{-8}$
0.3	0.09	3.56716 × ${10}^{-7}$	1.85430 × ${10}^{-7}$
0.4	0.16	7.48135 × ${10}^{-7}$	3.84891 × ${10}^{-7}$
0.5	0.25	1.45995 × ${10}^{-6}$	6.85008 × ${10}^{-7}$
0.6	0.36	2.32864 × ${10}^{-6}$	1.10671 × ${10}^{-6}$
0.7	0.49	3.47190 × ${10}^{-6}$	1.73964 × ${10}^{-6}$
0.8	0.64	5.19781 × ${10}^{-6}$	2.46654 × ${10}^{-6}$
0.9	0.81	7.05227 × ${10}^{-6}$	3.56779 × ${10}^{-6}$
1.0	1.00	9.28981 × ${10}^{-6}$	4.45749 × ${10}^{-6}$