## 5 Quadrature Methods (QM)  
In this Section, we explore the use of some of the numerical integration techniques to approximate the integral operator in (1) and to thus obtain a semi-discrete equation of the kind (2). A numerical integration or numerical quadrature formula may be written in the form  
\[\int_{a}^{b}y(x)dx=\sum_{j=1}^{n+1}w_{j}y(x_{j})+E_{n}(y), \tag{31}\]  
where \(y(x)\in C[a,b]\). The abscissas, usually equally spaced points, \(x_{j}\), \(j=1,2,\ldots,n+1\), and the weights \(w_{j}\), \(j=1,2,\ldots,n+1\), are determined only by the quadrature rule that we apply and do not depend on any way upon the integrand \(y(x)\). \(E_{n}(y)\) denotes the quadrature error which depends upon \(a,b,n\) and the value of a higher-order derivative of \(y(x)\) at some point between \(a\) and \(b\)[22]. Using (31), we may express the definite integral in (1) as  
\[\int_{a}^{b}K(x,t)u(t)dt=\sum_{j=1}^{n+1}w_{j}K(x,t_{j})u(t_{j})+E_{n}(K,u), \tag{32}\]  
where \(\{t_{j}\}\) is a set of \(n+1\) points in \([a,b]\), \(\{w_{j}\}\) is a specific set of positive weights not depending on \(x\) and \(\{t_{j}\}\), and \(E_{n}(K,u)\) is an error function which depends upon \(x\) as well as \(a,b,n\) and the values of higher-order derivatives of \(K(x,t)\) and \(u(t)\) with respect to \(t\) at some point between \(a\) and \(b\). Substituting (32) into (1), we find  
\[u(x)-\lambda\sum_{j=1}^{n+1}w_{j}K(x,t_{j})u(t_{j})-\lambda E_{n}(K,u)=f(x), \quad x\in[a,b].\]After disregarding the error term, we obtain the semi-discrete equation  
\[\tilde{u}_{n}(x)-\lambda\sum_{j=1}^{n+1}w_{j}K(x,t_{j})\tilde{u}_{n}(t_{j})=f(x), \quad x\in[a,b], \tag{33}\]  
where \(\tilde{u}_{n}\) denotes an approximate solution of \(u\). By specifying the vectors  
\[G = \lambda\big{(} \begin{array}{ccccc}g_{1}&g_{2}&\ldots&g_{n+1} \end{array} \big{)}\] \[= \lambda\big{(} \begin{array}{ccccc}w_{1}K(x,t_{1})&w_{2}K(x,t_{2 })&\ldots&w_{n+1}K(x,t_{n+1})\end{array} \big{)},\]  
and  
\[\Psi(\tilde{u}_{n})=\left( \begin{array}{c}\Psi_{1}(\tilde{u}_{n})\\ \Psi_{2}(\tilde{u}_{n})\\ \vdots\\ \Psi_{n+1}(\tilde{u}_{n})\end{array} \right),\quad\Psi_{j}(\tilde{u}_{n})= \tilde{u}_{n}(t_{j}),\quad j=1,2,\ldots,n+1,\]  
Equation (33) may be recast into symbolic form  
\[T_{q}\tilde{u}_{n}=\tilde{u}_{n}-G\Psi(\tilde{u}_{n})=f, \tag{34}\]  
where the operator \(T_{q}:X\to X\) and \(X=C[a,b]\). This equation is of the type (3) and its unique solution for the entire interval \([a,b]\) is  
\[\tilde{u}_{n}=T_{q}^{-1}f=f+G[\mathbf{I}_{n+1}-\Psi(G)]^{-1}\Psi(f), \tag{35}\]  
by means of Theorem 1. We observe that in (35), the evaluation of \(\Psi(G)\) and \(\Psi(f)\) consists merely of the computation of the functions \(g_{j}(x)\), \(j=1,2,\ldots,n+1\), and \(f(x)\) at the quadrature points. Moreover, Formula (35) corresponds to what is known as the natural interpolation form of Nystrom, which is one of the most efficient methods for computing accurate approximate values of the true solution in the entire interval \([a,b]\) from its approximate values at a set of nodes in \([a,b]\); see [5; 6] for more details. As an alternative to this, we may use other interpolating schemes to construct an approximate solution of specific type throughout the interval \([a,b]\). We consider below two such cases where the functions \(\{g_{j}(x)\}\) and \(f(x)\) are replaced by other, simpler functions, such as Taylor series and polynomials. Let us approximate each of \(g_{j}(x)\), \(j=1,2,\ldots,n+1\) and \(f(x)\) by partial sums of Taylor series as in (20) and (18), respectively. Then, by means of (21), Equation (34) is carried to  
\[T_{qs}\tilde{u}_{n}=\tilde{u}_{n}-G_{r}\Psi(\tilde{u}_{n})=f_{r}, \tag{36}\]  
where the operator \(T_{qs}:X\to X\) and \(X=C[a,b]\). Analogously, we may approximate each of \(g_{j}(x)\), \(j=1,2,\ldots,n+1\) and \(f(x)\) by interpolating polynomials as in (27) and (26), respectively.