First, we rewrite equations (17), (21), (25), and (29) in the following form:
$$ F_{j}(s)+G_{j}(s)=g(s),\quad j=1,2,3,4, $$
(30)
where
$$\begin{aligned} & F_{1}(s)= \int_{-1}^{1} \sqrt{1-t^{2}} \frac{x(t)-x(s)}{t-s}\,dt, \\ &G_{1}(s)=-\pi s x(s), \quad\mbox{for Case }(\text{i}), \\ &F_{2}(s)= \int_{-1}^{1} \frac{1}{\sqrt{1-t^{2}}} \frac {x(t)-x(s)}{t-s}\,dt, \\ \begin{aligned} &G_{2}(s)=0, \quad\mbox{for Case }(\text{ii}), \\ &F_{3}(s)=\frac{1}{1+s} \biggl( \int_{-1}^{1}\sqrt{1-t^{2}} \frac {x(t)-x(s)}{t-s}\,dt- \int_{-1}^{1}\sqrt{1-t^{2}} \frac{x(t)}{1+t}\,dt \biggr), \end{aligned} \\ &G_{3}(s)=-\frac{\pi s x(s)}{1+s}, \quad\mbox{for Case }(\text{iii}), \\ &F_{4}(s)=\frac{1}{1-s} \biggl( \int_{-1}^{1}\sqrt{1-t^{2}} \frac {x(t)-x(s)}{t-s}\,dt+ \int_{-1}^{1}\sqrt{1-t^{2}} \frac{x(t)}{1-t}\,dt \biggr), \\ &G_{4}(s)=-\frac{\pi s x(s)}{1-s}, \quad\mbox{for Case }(\text{iv}), \end{aligned}$$
(31)
and \(\frac{x(t)-x(s)}{t-s}=x'(s)\) while \(t=s\). According to equations (31) and by placing the collocation points \(s_{m}\) defined by
$$ s_{m}=-1+\frac{2}{n+2}(m+1),\quad m=0,1,\dots,n, $$
(32)
into (30), we get
$$ F_{j}(s_{m})+G_{j}(s_{m})=g(s_{m}),\quad j=1,2,3,4. $$
(33)
Using (11) and (31), we obtain
$$ \begin{aligned}& F_{1}(s_{m})\simeq \sum_{i=0}^{n}x_{1,i} \int_{-1}^{1} \sqrt{1-t^{2}} \frac {B_{i,n}(t)-B_{i,n}(s_{m})}{t-s_{m}}\,dt, \quad\mbox{for Case }(\text{i}), \\ &F_{2}(s_{m})\simeq\sum_{i=0}^{n}x_{2,i} \int_{-1}^{1} \frac{1}{\sqrt {1-t^{2}}} \frac{B_{i,n}(t)-B_{i,n}(s_{m})}{t-s_{m}}\,dt,\quad \mbox{for Case }(\text{ii}), \\ &F_{3}(s_{m})\simeq\frac{1}{1+s_{m}} \Biggl( \sum _{i=0}^{n}x_{3,i} \int _{-1}^{1}\sqrt{1-t^{2}} \frac{B_{i,n}(t)-B_{i,n}(s_{m})}{t-s_{m}}\,dt \\ &\phantom{F_{3}(s_{m})\simeq}{}- \sum_{i=0}^{n}x_{3,i} \int_{-1}^{1}\sqrt{1-t^{2}} \frac {B_{i,n}(t)}{1+t}\,dt \Biggr), \quad\mbox{for Case }(\text{iii}), \\ &F_{4}(s_{m})\simeq\frac{1}{1-s_{m}} \Biggl( \sum _{i=0}^{n}x_{4,i} \int _{-1}^{1}\sqrt{1-t^{2}} \frac{B_{i,n}(t)-B_{i,n}(s_{m})}{t-s_{m}}\,dt \\ &\phantom{F_{4}(s_{m})\simeq}{}+ \sum_{i=0}^{n}x_{4,i} \int_{-1}^{1}\sqrt{1-t^{2}} \frac {B_{i,n}(t)}{1-t}\,dt \Biggr), \quad\mbox{for Case }(\text{iv}) \end{aligned} $$
(34)
and
$$\begin{aligned} & G_{1}(s_{m})\simeq-\sum _{i=0}^{n}x_{1,i} B_{i,n}(s_{m}) \pi s_{m}, \quad\mbox{for Case }(\text{i}), \\ &G_{2}(s_{m})=0, \quad\mbox{for Case }(\text{ii}), \\ &G_{3}(s_{m})\simeq-\sum_{i=0}^{n}x_{3,i} B_{i,n}(s_{m})\frac{\pi s_{m} }{1+s_{m}}, \quad\mbox{for Case }(\text{iii}), \\ &G_{4}(s_{m})\simeq-\sum_{i=0}^{n}x_{4,i} B_{i,n}(s_{m})\frac{\pi s_{m} }{1-s_{m}}, \quad \mbox{for Case }(\text{iv}). \end{aligned}$$
We use the Gauss-Chebyshev quadrature rule of the first kind for computing the integral part in Case (ii) and select the Gauss-Chebyshev quadrature rule of the second kind for computing the integral part in the other cases. So, we can rewrite (33) for all cases as follows:
$$ \begin{aligned} & \sum_{i=0}^{n} \Biggl[\sum_{k=1}^{N} \omega_{k} \frac {B_{i,n}(t_{k})-B_{i,n}(s_{m})}{t_{k}-s_{m}}- B_{i,n}(s_{m})\pi s_{m} \Biggr]x_{1,i}=g(s_{m}), \quad\mbox{for Case }(\text{i}), \\ & \sum_{i=0}^{n} \Biggl[\sum _{k=1}^{N} \omega_{k} \frac {B_{i,n}(t_{k})-B_{i,n}(s_{m})}{t_{k}-s_{m}} \Biggr]x_{2,i}=g(s_{m}),\quad \mbox{for Case }(\text{ii}), \\ & \sum_{i=0}^{n} \biggl[ \frac{ \sum_{k=1}^{N} \omega_{k} \frac {B_{i,n}(t_{k})-B_{i,n}(s_{m})}{t_{k}-s_{m}}- \sum_{k=1}^{N} \omega_{k} \frac {B_{i,n}(t_{k})}{1+t_{k}} }{1+s_{m}}-\frac{\pi s_{m} B_{i,n}(s_{m})}{1+s_{m}} \biggr]x_{3,i}=g(s_{m}), \\ &\quad\mbox{for Case }(\text{iii}), \\ & \sum_{i=0}^{n} \biggl[ \frac{ \sum_{k=1}^{N} \omega_{k} \frac {B_{i,n}(t_{k})-B_{i,n}(s_{m})}{t_{k}-s_{m}}- \sum_{k=1}^{N} \omega_{k} \frac {B_{i,n}(t_{k})}{1-t_{k}} }{1-s_{m}}-\frac{\pi s_{m} B_{i,n}(s_{m})}{1-s_{m}} \biggr]x_{4,i}=g(s_{m}), \\ &\quad\mbox{for Case }(\text{iv}), \end{aligned} $$
(35)
where
$$ \begin{aligned}& t_{k}=\cos \biggl( \frac{k\pi}{N+1} \biggr), \\ &\omega_{k}=\frac{\pi }{N+1}{ \biggl(\sin \biggl( \frac{k\pi}{N+1} \biggr) \biggr)}^{2}\quad \mbox{for Cases } \text{(i), (iii), (iv)}, \\ & t_{k}=\cos \biggl(\frac{(2k-1)\pi}{2N} \biggr),\\ & \omega_{k}= \frac{\pi }{N},\quad \mbox{for Case }(\text{ii}), k=1,2,\dots,N. \end{aligned} $$
(36)
For simplicity, we write equations (35) as follows:
$$ \sum_{i=0}^{n} \bigl[ \mathcal{F}_{j}(s_{m})+\mathcal{G}_{j}(s_{m}) \bigr]x_{j,i}=g(s_{m}),\quad j=1,2,3,4. $$
(37)
Hence, the main matrix form (37) corresponding to all cases of (2) can be written separately in the form
$$ \mathbf{A}_{j}\mathbf{X}_{j}=\mathbf{G},\quad j=1,2,3,4, $$
(38)
where
$$\begin{aligned} &[\mathbf{A}_{j}]_{(m+1)(i+1)}= \mathcal{F}_{j}(s_{m})+\mathcal{G}_{j}(s_{m}),\quad m,i=0,1,\dots,n, \\ &\mathbf{X}_{j}=[x_{j,0},x_{j,1}, \dots,x_{j,n}]^{T},\quad j=1,2,3,4, \end{aligned}$$
and
$$\mathbf{G}=\bigl[g(s_{0}),g(s_{1}),\dots,g(s_{n}) \bigr]^{T},\quad \mbox{for Cases }(\text{i}), (\text{ii}), (\text{iii}), (\text{iv}). $$
After solving equations (38) for Cases (i), (ii), (iii) and (iv), the unknown coefficients \(x_{j,i}\) are determined and we can approximate the solutions of (17), (21), (25) and (29) with substituting \(x_{j,i}\), \(i=0,1,\dots ,n\)
\(j=1,2,3,4\) in (11). So, the approximate solutions for (2) in all cases follow:
$$ \begin{aligned} &y_{n}(s)=\frac{1-s^{2}}{\sqrt{1-s^{2}}}x_{n}(s)= \frac{1-s^{2}}{\sqrt {1-s^{2}}}\sum_{i=0}^{n} x_{1,i} B_{i,n}(s), \quad\mbox{for Case }(\text{i}), \\ &y_{n}(s)=\frac{1}{\sqrt{1-s^{2}}}x_{n}(s)=\frac{1}{\sqrt{1-s^{2}}}\sum _{i=0}^{n} x_{2,i} B_{i,n}(s), \quad\mbox{for Case }(\text{ii}), \\ &y_{n}(s)=\sqrt{\frac{1-s}{1+s}}x_{n}(s)=\sqrt{ \frac{1-s}{1+s}}\sum_{i=0}^{n} x_{3,i} B_{i,n}(s), \quad\mbox{for Case }(\text{iii}), \\ &y_{n}(s)=\sqrt{\frac{1+s}{1-s}}x_{n}(s)=\sqrt{ \frac{1+s}{1-s}}\sum_{i=0}^{n} x_{4,i} B_{i,n}(s), \quad\mbox{for Case } (\text{iv}). \end{aligned} $$
(39)