Direct regularization for system of integral-algebraic equations of index-1

Figures & data

Figure 1. Error behaviours of Test Prob. 1 for different values of N .

Figure 2. The effect of $\mathit{\alpha }$ on the number of iteration m for Test Prob. 1 with $N=32$.

Figure 3. Exact and approximate solutions of Test Prob. 1 for $N=8,32$.

Figure 4. Error behaviours of Test Prob. 2 for different values of N .

Figure 5. The effect of $\mathit{\alpha }$ on the number of iteration m for Test Prob. 2 with $N=32$.

Figure 6. Exact and approximate solutions of Test Prob. 2 for $N=8,32$.

Figure 7. Error behaviours of Test Prob. 3 for different values of N .

Figure 8. The effect of $\mathit{\alpha }$ on the number of iteration m for Test Prob. 3 with $N=32$.

Figure 9. Exact and approximate solutions of Test Prob. 3 for $N=8,32$.

Table

Test Prob.	K(t,s)	Exact solution
2*1	2*$\left(\begin{array}{cc}t+s& t^2+s^2\\ s-t^2& t+s+1\\ \end{array}\right)$	$y(t)=sint$
		$z(t)=cost$
2*2	2*$\left(\begin{array}{cc}{t}^3+s+1& cos3s+1\\ t+s+2& sin3s+2\\ \end{array}\right)$	$y(t)=cost$
		$z(t)=sin3t$
2*3	2*$\left(\begin{array}{cc}1+t-s& e^{t-s}\\ 1& e^{2t-s}\\ \end{array}\right)$	$y(t)=t{e}^{-t}$
		$z(t)=cost$

Table 1. Numerical results of Test Prob. 1 for different values of N, m and $\mathit{\alpha }$.

N	$\mathit{\alpha }$	m	$\Vert y_{\mathit{\alpha }}^{\mathit{\delta }}{-\overline{y}\Vert }_2$	$\Vert z_{\mathit{\alpha }}^{\mathit{\delta }}{-\overline{z}\Vert }_2$	Time (s)
2*4	0.05	5	9.21E$-$02	7.56E$-$01	0.15
	0.025	4	6.45E$-$02	5.48E$-$02	0.14
2*8	0.025	30	3.11E$-$02	3.27E$-$02	1.93
	0.01	6	3.11E$-$02	2.88E$-$02	0.40
2*16	0.01	28	1.54E$-$02	1.67E$-$02	3.73
	0.005	12	1.54E$-$02	1.55E$-$02	1.60
3*32	2*0.005	50	1.06E$-$02	6.45E$-$01	15.67
		61	7.69E$-$03	7.97E$-$03	19.48
	0.001	10	7.70E$-$03	7.98E$-$03	3.14

Table 2. Numerical results of Test Prob. 2 for different values of N, m and $\mathit{\alpha }$.

N	$\mathit{\alpha }$	m	$\Vert y_{\mathit{\alpha }}^{\mathit{\delta }}{-\overline{y}\Vert }_2$	$\Vert z_{\mathit{\alpha }}^{\mathit{\delta }}{-\overline{z}\Vert }_2$	Time (s)
2*4	2*0.05	4	3.90E$-$02	1.87E$-$01	0.14
		7	3.90E$-$02	1.79E$-$01	0.26
2*8	2*0.025	2	1.89E$-$02	1.03E$-$01	0.12
		7	1.90E$-$02	8.25E$-$02	0.46
3*16	0.025	65	9.44E$-$03	4.22E$-$02	9.48
	2*0.01	5	9.44E$-$03	2.87E$-$01	0.68
		15	9.44E$-$03	4.26E$-$02	1.93
3*32	2*0.01	108	4.71E$-$03	1.43E$-$01	59.01
		130	4.71E$-$03	2.12E$-$02	88.85
	0.001	6	4.71E$-$03	2.12E$-$02	1.79

Table 3. Numerical results of Test Prob. 3 for different values of N, m and $\mathit{\alpha }$.

N	$\mathit{\alpha }$	m	$\Vert y_{\mathit{\alpha }}^{\mathit{\delta }}{-\overline{y}\Vert }_2$	$\Vert z_{\mathit{\alpha }}^{\mathit{\delta }}{-\overline{z}\Vert }_2$	Time (s)
2*4	2*0.05	6	5.58E$-$02	5.82E$-$01	0.26
		10	3.74E$-$02	4.60E$-$02	0.37
4*8	2*0.025	18	2.65E$-$02	6.96E$-$01	1.42
		30	1.72E$-$02	2.37E$-$02	2.35
	2*0.01	5	1.72E$-$02	1.34E$-$01	0.35
		8	1.72E$-$02	2.28E$-$02	0.54
4*16	2*0.01	27	1.37E$-$02	9.69E$-$01	5.31
		34	8.44E$-$03	1.13E$-$02	6.64
	2*0.005	10	8.68E$-$03	2.59E$-$01	1.85
		14	8.44E$-$03	1.11E$-$02	2.60
4*32	2*0.005	75	4.51E$-$03	3.53E$-$01	28.96
		90	4.20E$-$03	5.73E$-$03	36.18
	2*0.001	5	4.29E$-$03	8.54E$-$01	1.62
		12	4.20E$-$03	5.72E$-$03	3.93