## 5 Numerical Aspects
### Generalized Integral Quadrature Method  
The GIQ method is a numerical tool that allows to evaluate integrals. Let us consider a smooth function \(f(x)\) defined in a closed domain \([a,b]\). In addition, let us assume that \(I_{T}\) points are placed to discretize the domain itself, so that one gets \(a=x_{1},x_{2},\ldots,x_{I_{T}-1},x_{I_{T}}=b\). As a consequence, \(I_{T}-1\) subdomains are defined. The continuous function \(f(x)\) can be approximated by the Lagrange polynomials of order \(I_{T}-1\), since \(I_{T}\) points are employed to discretize the domain. The integral of the approximated function in the closed interval \(\big{[}x_{i},x_{j}\big{]}\) is defined as follows  
\[\int\limits_{x_{i}}^{x_{j}}f(x)dx=\sum\limits_{k=1}^{I_{T}}w_{k}^{ij}f(x_{k}) \tag{67}\]  
In other words, the numerical integral of \(f(x)\) is a weighted linear sum of the values that the function \(f(x)\) assumes in each point of the reference domain. It should be noted that this procedure involves the values that the function assumes in the points placed outside the integration domain. As in the GDQ method, \(w_{k}^{ij}\) are the weighting coefficients required for the integration, which can be computed as shown below. Let us assume that the function \(f(x)\) is approximated by a polynomial expansion as  
\[f(x)=a_{1}+a_{2}x+\cdots+a_{I_{T}}x^{I_{T}-1} \tag{68}\]  
in which \(a_{i}\), for \(i=1,2,...,I_{T}\), denote the arbitrary constants of the polynomial. An auxiliary function \(F(x)\) must be introduced so that  
\[f(x)=\frac{dF(x)}{dx} \tag{69}\]  
Due to Equation (68), the function \(F(x)\) can be defined as  
\[F(x)=\int\limits_{c}^{x}f(\lambda)d\lambda+u(c)=I_{f}(x,c)+u(c) \tag{70}\]  
in which \(I_{f}(x,c)\) is the integrand function that assumes the following aspect  
\[I_{f}(x,c)=x\bigg{(}a_{1}+\frac{a_{2}}{2}x+\cdots+\frac{a_{I_{T}}}{I_{T}}x^{I_ {T}-1}\bigg{)}-c\bigg{(}a_{1}+\frac{a_{2}}{2}c+\cdots+\frac{a_{I_{T}}}{I_{T}}c ^{I_{T}-1}\bigg{)} \tag{71}\]On the other hand, \(c\) denotes an arbitrary constant. Due to the Weierstrass theorem and the properties of a linear vector space, the expression below represents an approximation of the function \(I_{f}(x,c)\)  
\[I_{f}(x,c)=\sum_{j=1}^{I_{T}}{(x-c)}l_{j}(x)d_{j} \tag{72}\]  
where \(l_{j}(x)\) specifies the Lagrange polynomials. The first derivative of \(I_{f}(x,c)\) with respect to \(x\) can be easily evaluated at the point \(x_{i}\)  
\[\left.\frac{dI_{f}(x,c)}{dx}\right|_{x_{i}}\ =\ I_{f}^{(1)}(x,c)\Big{|}_{x_{i}}= \sum_{j=1}^{I_{T}}{d_{j}\Big{(}l_{j}(x_{i})+(x_{i}-c)l_{j}^{(1)}(x_{i})\Big{)}} \tag{73}\]  
where \(l_{j}^{(1)}(x_{i})\) stands for the first-order derivative with respect to \(x\) of the Lagrange polynomials computed at \(x_{i}\). It should be noted that \(l_{j}^{(1)}(x_{i})\) stands for the first-order derivative with respect to \(x\) of the Lagrange polynomials computed at \(x_{i}\) and coincides with the weighting coefficients of the first order derivative \(\varsigma_{ij}^{(1)}\) given by the GDQ method [124]. By applying the differential quadrature law, Equation (73) becomes  
\[\left.\frac{dI_{f}(x,c)}{dx}\right|_{x_{i}}=\left.I_{f}^{(1)}(x,c)\right|_{x_ {i}}=\sum_{j=1}^{I_{T}}{\overline{\varsigma}}_{ij}^{(1)}I_{f}(x_{j},c)=\sum_{ j=1}^{I_{T}}{\overline{\varsigma}}_{ij}^{(1)}\sum_{k=1}^{I_{T}}{d_{k}\big{(}x_{j}-c \big{)}l_{k}\big{(}x_{j}\big{)}} \tag{74}\]  
in which the terms \(l_{k}\big{(}x_{j}\big{)}=\delta_{kj}\) represent the Kronecker delta. The following results is achieved by comparing Equations (73) and (74)  
\[\left.\frac{dI_{f}(x,c)}{dx}\right|_{x_{i}}=\left.I_{f}^{(1)}(x,c)\right|_{x_ {i}}=\sum_{j=1}^{I_{T}}{\overline{\varsigma}}_{ij}^{(1)}\big{(}x_{j}-c\big{)} d_{j} \tag{75}\]  
where the weighting coefficients assume the form  
\[{\overline{\varsigma}}_{ij}^{(1)}=\frac{x_{i}-c}{x_{j}-c}\varsigma_{ij}^{(1)} \tag{76}\]  
for \(i\
eq j\), and  
\[f(x)=a_{1}+a_{2}x+\cdots+a_{I_{T}}x^{I_{T}-1} \tag{68}\]0  
for \(i=j\). The arbitrary constant \(c\) must be set equal to \(c=x_{I_{T}}+10^{-10}\) due to accuracy and stability reasons [124].