# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 5 Optimal Coefficients of Weighted Quadrature Formulas
: _In the Sobolev space \(L^{(2)}_{2}(0,1)\), the optimal quadrature formula is the following formula_  
\[\int_{0}^{1}p(x)\varphi(x)dx\cong\sum_{\beta=0}^{N}\left(\overset{\circ}{C}^{ (0)}[\beta]\varphi[\beta]+\overset{\circ}{C}^{(1)}[\beta]\varphi^{\prime}[ \beta]\right).\]Here \(\overset{\circ}{\text{C}}^{(0)}[\beta]\) are determined from Theorem 5, and the optimal coefficients \(\overset{\circ}{\text{C}}^{(1)}[\beta]\) are computed by the following formulas below:_  
\[\overset{\circ}{\text{C}}^{(1)}[0]=-\frac{h^{-1}}{2}\int_{0}^{h}p(x)(x-h)xdx,\]  
\[\overset{\circ}{\text{C}}^{(1)}[\beta]=\frac{h^{-1}}{2}\int_{h(\beta-1)}^{h( \beta+1)}p(x)[(x-h\beta)^{2}sign(h\beta-x)+h(x-h\beta)]dx,\beta=\overline{1,N-1},\]  
\[\overset{\circ}{\text{C}}^{(1)}[N]=\frac{h^{-1}}{2}\int_{1-h}^{1}p(x)(x-1+h)(x -1)dx.\]  
**Theorem 7**. : _The optimal coefficients \(\overset{\circ}{\text{C}}^{(2)}[\beta]\) of the quadrature formula of the form_  
\[\int_{0}^{1}p(x)\varphi(x)dx\cong\sum_{\beta=0}^{N}\left(\overset{\circ}{ \text{C}}^{(0)}[\beta]\varphi[\beta]+\overset{\circ}{\text{C}}^{(1)}[\beta] \varphi^{\prime}[\beta]+\overset{\circ}{\text{C}}^{(2)}[\beta]\varphi^{\prime \prime}[\beta]\right),\]  
_in the space \(L_{2}^{(3)}(0,1)\) are defined by the formulas_  
\[\overset{\circ}{\text{C}}^{(2)}[0]=\frac{h^{-1}}{12}\int_{0}^{h}p(x)x(h-x)(2 x-h)xdx,\]  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]0  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]1  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]2  
Application of the Quadrature Formula with Derivatives to Linear Fredholm Equations of the Second Kind  
Consider the following linear Fredholm equation of the second kind  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]3  
where \(K(x,s)\) is the kernel of the integral equation, \(f(x)\) is the right-hand side, \(\lambda\) is the parameter of the integral equation and \(y(x)\) is an unknown function to be determined. Various numerical techniques--such as collocation methods, projection methods, and Galerkin methods--have been proposed for approximating the solution of Equation (50). These methods have been thoroughly studied with respect to their stability and convergence within appropriate function spaces, taking into account the smoothness characteristics of both the kernel \(K\) and the right-hand side function \(f\) (see, for example, [49; 50; 51; 52; 53; 54; 55; 56; 57]). To solve Equation (50) we apply the quadrature Formula (26) and pass the difference grid on the argument x  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]4  
Here \(\mathcal{C}^{(k)}_{\beta}\) are the optimal coefficients of the quadrature formula, \(f_{i}=f(x_{i}),y_{i}=y(x_{i})\), \(y^{(k)}_{i}=y^{(k)}(x_{i}),x_{i}=i*h,i=\overline{0,N},h\) is grid spacing. In the system of Equation (51) the number of equations is \(N+1\), and the number of unknowns is \((N+1)*m\), i.e., in addition to the unknown function, its derivatives at nodal points participate in the system of equations. To solve this problem, we differentiate Equation (50) \(m-1\) times by the argument \(x\), and we have  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]5  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]6  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]7  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]8  
Now applying the quadrature formula to system (52), we obtain  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]9  
\[\overset{\circ}{C}^{(0)}[\beta]=h^{-1}\int_{h(\beta-1)}^{h(\beta+1)}p(x)[(h \beta-x)sign(x-h\beta)+h]dx,\beta=\overline{1,N-1},\]0  
\[\overset{\circ}{C}^{(0)}[\beta]=h^{-1}\int_{h(\beta-1)}^{h(\beta+1)}p(x)[(h \beta-x)sign(x-h\beta)+h]dx,\beta=\overline{1,N-1},\]1  
\[\overset{\circ}{C}^{(0)}[\beta]=h^{-1}\int_{h(\beta-1)}^{h(\beta+1)}p(x)[(h \beta-x)sign(x-h\beta)+h]dx,\beta=\overline{1,N-1},\]2  
Thus, we have a system of linear algebraic equations with respect \(y^{(k)}_{i}(i=\overline{0,N},\)\(k=\overline{0,m-1})\).