Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications

Abstract

The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. It is known that the numerical integration of definite integrals is of great importance in basic and applied sciences. In this paper we consider the optimization problem of weighted quadrature formulas with derivatives in Sobolev space. Using the extremal function, the square of the norm of the error functional of the considered quadrature formula is calculated. Then, minimizing this norm by coefficients, we obtain a system to find the optimal coefficients of this quadrature formula. The uniqueness of solutions of this system is proved, and an algorithm for solving this system is given. The proposed algorithm is used to obtain the optimal coefficients of the derivative weight quadrature formulas. It should be noted that the optimal weighted quadrature formulas constructed in this work are optimal for the approximate calculation of regular, singular, fractional and strongly oscillating integrals. The constructed optimal quadrature formulas are applied to the approximate solution of linear Fredholm integral equations of the second kind. Finally, the numerical results are compared with the known results of other authors.

1. Introduction

An important practical goal of computational mathematics is to create the best, i.e., the fastest and cheapest ways of solving mathematical problems. In short, the optimization of computational algorithms. The optimization of computational algorithms is well demonstrated by examples of the construction of cubature and quadrature formulas on the functional formulation. In this formulation, we consider functions $\phi \left(x\right)$ belonging to some Banach space B. It is assumed that this space is nested in the space of continuous functions defined in the domain $\Omega$. The integral of the function $\phi \left(x\right)$ with the weight function $p\left(x\right)$ over the region $\Omega$

$${\int }_{\Omega }p\left(x\right)\phi \left(x\right)dx$$

is a linear functional in B. Its approximate expression is.

$$\sum _{k=1}^N{C}_k\phi \left(x^k\right)$$

will be another linear functional. The error functional [1,2] of the cubature formula will also be linear

$$(\ell ,\phi )={\int }_{\Omega }p\left(x\right)\phi \left(x\right)dx-\sum _{k=1}^N{C}_k\phi \left(x^{\left(k\right)}\right)=$$

$$={\int }_{\Omega }{\epsilon }_{\omega }\left(x\right)p\left(x\right)-\sum _{k=1}^N{C}_k\delta (x-x^{\left(k\right)})]\phi \left(x\right)dx.$$

(1)

The problem of constructing a cubature formula

$${\int }_{\Omega }p\left(x\right)\Omega \left(x\right)dx\cong \sum _{k=1}^N{C}_k\phi \left(x^{\left(k\right)}\right)$$

(2)

in the functional formulation consists of finding such a functional (1) whose norm in the space $B^{*}$ is minimal.

Studies of optimal and asymptotic cubature formulas are found in [1,2,3,4,5,6,7,8,9,10,11]. Optimization studies of quadrature formulas are presented in [12,13,14].

Currently, there are various methods for constructing optimal approximate integration formulas: the spline method, the $\phi$ function method and the Sobolev method.

In recent years, significant progress has been made in developing optimal quadrature formulas and analyzing their error estimates for the approximate evaluation of regular functions [15,16,17,18,19,20,21,22,23,24], singular [25,26], fractional [27] and integrals from rapidly oscillating functions [28] using the Sobolev method.

In this paper, the construction of composite optimal quadrature formulas with weight in the Sobolev space is studied using the variational method. Here, the square of the norm of the error functional of composite quadrature formulas with weight function is computed using the extremal function. By minimizing the given norm concerning the coefficients, a system of algebraic equations is derived. The uniqueness of the solution to this system is established. This approach allows for the optimal coefficients of quadrature formulas involving a weight function to be determined.

In the case where the weight function is equal to one, the coefficients of the well-known Euler–Maclorean quadrature formula are obtained from the general formula for the optimal coefficients.

The developed optimal quadrature formula is applied to obtain approximate solutions of certain linear Fredholm integral equations of the second kind, and the results are compared with those presented in [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].

2. Compound Quadrature Formulas of Hermite Type

Let us consider quadrature formulas of the form

$${\int }_0^{1}p\left(x\right)\phi \left(x\right)dx\cong \sum _{\beta =0}^N\sum _{\nu =0}^t{C}_{\beta }^{\left(\nu \right)}{\phi }^{\left(\nu \right)}\left(x_{\beta }\right)$$

(3)

in the space $L_2^{\left(m\right)}(0,1)$. Here $L_2^{\left(m\right)}(0,1)$ is the space of functions whose $m^{th}$ generalized derivative sums to square on the interval [0, 1]; $p\left(x\right)$ is the weight function whose

$${\int }_0^{1}p\left(x\right)dx<\infty ;$$

$C_{\beta }^{\left(\nu \right)}$ are coefficients; $x_{\beta }$ are nodes of quadrature formulas; and $\beta =\overline{0,N},\nu =\overline{0,t},t=\overline{0,m-1}.$ Here, the integral is considered to be regular, singular, fractional and strongly oscillating.

The error of the quadrature formula is the difference between the integral and the sum

$$({\ell }_N^{\left(t\right)},\phi )={\int }_0^{1}p\left(x\right)\phi \left(x\right)dx-\sum _{\beta =0}^N\sum _{\nu =0}^t{C}_{\beta }^{\left(\nu \right)}{\phi }^{\left(\nu \right)}\left(x_{\beta }\right)={\int }_0^1{\ell }_N^{\left(t\right)}\left(x\right)\phi \left(x\right)dx,$$

where

$${\ell }_N^{\left(t\right)}\left(x\right)={\mathcal{E}}_{[0,1]}\left(x\right)p\left(x\right)-\sum _{\beta =0}^N\sum _{\nu =0}^t{(-1)}^{\nu }{C}_{\beta }^{\left(\nu \right)}{\delta }^{\left(\nu \right)}(x-x_{\beta }),$$

(4)

${\mathcal{E}}_{[0,1]}$ is the characteristic function of the interval [0, 1], $\delta \left(x\right)$ is Dirac’s delta function, ${\ell }_N^{\left(t\right)}$ is the error functional of the quadrature Formula (3).

The functional ${\ell }_N^{\left(t\right)}\left(x\right)$ of the form (4) is defined in the space $L_2^{\left(m\right)}(0,1)$, i.e., this functional belongs to the conjugate space $L_2^{\left(m\right)*}(0,1)$, then we have

$$({\ell }_N^{\left(t\right)},x^{\alpha })=0,\alpha =0,1,\dots ,m-1.$$

(5)

The problem of constructing an optimal quadrature formula of the form (3) with the error functional (4) in the space $L_2^{\left(m\right)}(0,1)$ consists of finding the value of

$$\Vert {\stackrel{\circ }{\ell }}_N^{\left(t\right)}|L_2^{\left(m\right)*}{(0,1)\Vert }^2=\underset{C_{\beta }^{\left(\nu \right)}}{\inf }|({\ell }_N^{\left(t\right)},{\psi }_{\ell })|$$

(6)

at fixed nodes $x_{\beta }$.

In Formula (6), ${\psi }_{\ell }\left(x\right)$ is the extremal function of the quadrature Formula (3) in the space $L_2^{\left(m\right)}(0,1)$.

Theorem 1.

The extremal function of the error functional ${\ell }_N^{\left(t\right)}\left(x\right)$ in the space $L_2^{\left(m\right)}(0,1)$ is of the form

$${\psi }_{\ell }\left(x\right)={(-1)}^m{\ell }_N^{\left(t\right)}\left(x\right)\ast G_m\left(x\right)+P_{m-1}\left(x\right),$$

(7)

where $G_m\left(x\right)$ is Green’s function of the operator ${\displaystyle \frac{d^{2m}}{d{x}^{2m}}}$, i.e.,

$$G_m\left(x\right)={\displaystyle \frac{x^{2m-1}sign\left(x\right)}{2(2m-1)!}},$$

$P_{m-1}\left(x\right)$ is some polynomial of degree $m-1$.

At $t=0$, i.e., for the functional

$$\ell \left(x\right)={\mathcal{E}}_{[0,1]}\left(x\right)-\sum _{\beta =0}^N{C}_{\beta }\delta (x-h\beta )$$

the extremal function was found in [1,2]. For any $0\le t\le m-1$, Theorem 1 is proved in [8].

Since $L_2^{\left(m\right)}(0,1)$ is a Hilbert space, the norm of the error functional ${\ell }_N^{\left(t\right)}$ is connected to the function ${\psi }_{\ell }\left(x\right)$ through the following relation:

$$\Vert {\ell }_N^{\left(t\right)}/L_2^{\left(m\right)*}{(0,1)\Vert }^2={\int }_0^1{\left({\psi }_{\ell }^{\left(m\right)}\left(x\right)\right)}^{2}dx.$$

(8)

In addition, there is an equality

$$\Vert {\ell }_N^{\left(t\right)}/L_2^{\left(m\right)*}{(0,1)\Vert }^2=(\ell ,{\psi }_{\ell }).$$

(9)

By substituting the extremal function given in Formula (7) into equality (9) and taking into account Expression (5), a series of calculations yields the squared norm of the error functional (4) corresponding to the quadrature Formula (3).

$$\Vert {\ell }_N^{\left(t\right)}/L_2^{\left(m\right)*}{(0,1)\Vert }^2={(-1)}^m[\sum _{\beta =0}^N\sum _{{\beta }^{\prime }=0}^N\sum _{\nu =0}^t\sum _{{\nu }^{\prime }=0}^t{C}_{\beta }^{\left(\nu \right)}{C}_{{\beta }^{\prime }}^{\left({\nu }^{\prime }\right)}{G}_m^{(\nu +{\nu }^{\prime })}(x_{\beta }-x_{{\beta }^{\prime }})-$$

$$-2\sum _{\beta =0}^N\sum _{\nu =0}^t{C}_{\beta }^{\left(\nu \right)}{\int }_0^{1}p\left(x\right)G_m^{\left(\nu \right)}(x-x_{\beta })dx+$$

$$+{\int }_0^1{\int }_0^{1}p\left(x\right)p\left(y\right)G_m(x-y)dxdy]\equiv F\left(C\right),$$

(10)

where

$$G_m^{\left(k\right)}\left(x\right)={\displaystyle \frac{x^{2m-1-k}sign\left(x\right)}{2(2m-1-k)!}}.$$

(11)

Recall that the coefficients $C_{\beta }^{\left(\nu \right)}$ in equality (10) must satisfy the system of linear equations

$$\sum _{\beta =0}^N\sum _{\nu =0}^t{C}_{\beta }^{\left(\nu \right)}{x}_{\beta }^{(k-\nu )}={\int }_0^{1}p\left(x\right)x^{k}dx,k=\overline{0,m-1}$$

(12)

which is equivalent to the accuracy conditions (5) of the quadrature formula for polynomials of degree less than m. In system (12)

$$x_{\beta }^{(k-\nu )}=\left\{\begin{array}{cc}{x}_{\beta }^{(k-\nu )},\hfill & \mathrm{if}k-\nu \ge 0,\hfill \\ 0,\hfill & \mathrm{if}k-\nu <0.\hfill \end{array}\right.$$

Now let us formulate the conditions of the quadratic function $F\left(C\right)$ on the set of vectors

$$C=(C_0^{\left(0\right)},C_1^{\left(0\right)},\dots ,C_N^{\left(0\right)},C_0^{\left(1\right)},C_1^{\left(1\right)},\dots ,C_N^{\left(1\right)},\dots ,C_0^{\left(t\right)},C_1^{\left(t\right)},\dots ,C_N^{\left(t\right)})$$

subject to relations (5). For this purpose, we apply the method of Lagrange indefinite multipliers.

Let us compose the Lagrange function

$$F(C,\lambda )=F\left(C\right)+2\sum _{k=0}^{m-1}{\lambda }_k({\ell }_N^{\left(t\right)},x^k).$$

Equating to zero the partial derivatives of $F(C,\lambda )$ by $C_{\beta }^{\left(\nu \right)}$ and ${\lambda }_k$, we obtain

$$\sum _{\beta =0}^N\sum _{\nu =0}^t{C}_{\beta }^{\left(\nu \right)}{G}_m^{(\nu +{\nu }^{\prime })}(x_{{\beta }^{\prime }}-x_{\beta })+\sum _{k=0}^{m-1}{\lambda }_k{x}_{{\beta }^{\prime }}^{(k-{\nu }^{\prime })}\times {\displaystyle \frac{k!}{(k-{\nu }^{\prime })!}}=f_{{\beta }^{\prime }}^{\left({\nu }^{\prime }\right)},$$

(13)

$${\nu }^{\prime }=\overline{0,t},{\beta }^{\prime }=\overline{0,N},$$

$$\sum _{\beta =0}^N\sum _{\nu =0}^N{\displaystyle \frac{k!}{(k-\nu )!}}{C}_{\beta }^{k-\nu }=g_k,k=\overline{0,m-1},$$

(14)

where

$$f_{{\beta }^{\prime }}^{\left({\nu }^{\prime }\right)}={\int }_0^{1}p\left(x\right)G_m^{\left({\nu }^{\prime }\right)}(x-x_{{\beta }^{\prime }})dx,{\nu }^{\prime }=\overline{0,t},{\beta }^{\prime }=\overline{0,N},$$

$$g_k={\int }_0^{1}p\left(x\right)x^{k}dx,k=\overline{0,m-1}.$$

In the system of Equations (13) and (14), the unknown variables are $C_{\beta }^{\left(\nu \right)}$ and ${\lambda }_k$. The solution to this system corresponds to a stationary point of the function $F(C,\lambda )$, which we denote by ${\stackrel{\circ }{C}}_{\beta }^{\left(\nu \right)}$ and ${\stackrel{\circ }{\lambda }}_k$. According to the theory of constrained extrema, there exists a sufficient condition ensuring that this solution represents a conditional minimum of $F\left(C\right)$ on the manifold defined by (5). This condition requires the associated quadratic form to be positive definite.

$$\varphi \left(C\right)=\sum _{\beta =0}^N\sum _{{\beta }^{\prime }=0}^N\sum _{\nu =0}^t\sum _{{\nu }^{\prime }=0}^t{\displaystyle \frac{{\partial }^{2}F(C,\lambda )}{\partial C_{\beta }^{\left(\nu \right)}\partial C_{{\beta }^{\prime }}^{\left({\nu }^{\prime }\right)}}}{C}_{\beta }^{\left(\nu \right)}{C}_{{\beta }^{\prime }}^{\left({\nu }^{\prime }\right)},$$

(15)

on the set of vectors $C_{\beta }^{\left(\nu \right)}$ obtaining the requirement

$$\sum _{\beta =0}^N\sum _{\nu =0}^t{C}_{\beta }^{\left(\nu \right)}{x}_{\beta }^{(k-\nu )}{\displaystyle \frac{k!}{(k-\nu )!}}=0,k=\overline{0,m-1}.$$

(16)

In matrix form system (16) has the form

$$SC=0.$$

(17)

We proceed to prove that in the considered case, the quadratic form (15) is positive definite.

Lemma 1.

For any nonzero vector C belonging to the subspace defined by $SC=0$, the function $\Phi \left(C\right)$ takes strictly positive values.*

Proof of Lemma 1.

From the definition of the Lagrange function $F(C,\lambda )$, and by utilizing equality (15), it directly follows that…

$$\Phi \left(C\right)=\sum _{\beta =0}^N\sum _{{\beta }^{\prime }=0}^N\sum _{\nu =0}^t\sum _{{\nu }^{\prime }=0}^t{C}_{\beta }^{\left(\nu \right)}{C}_{{\beta }^{\prime }}^{\left({\nu }^{\prime }\right)}{G}_m^{(\nu +{\nu }^{\prime })}(x_{\beta }-x_{{\beta }^{\prime }}).$$

(18)

Consider the following functional

$$\mu \left(C\right)=\sum _{\beta =0}^N\sum _{\nu =0}^t{(-1)}^{\nu }{C}_{\beta }^{\left(\nu \right)}{\delta }^{\left(\nu \right)}(x-x_{\beta }).$$

(19)

It is known that, by condition (6), this functional belongs to $L_2^{\left(m\right)*}$, i.e., $\mu \left(C\right)\in L_2^{\left(m\right)*}(0,1)$.

For this functional, there corresponds an extremal function ${\psi }_{\mu }\left(x\right)\in L_2^{\left(m\right)}(0,1)$, which is a solution of Equation

$${\displaystyle \frac{d^{2m}}{d{x}^{2m}}}{\psi }_{\mu }\left(x\right)={(-1)}^m{\ell }_{\mu }\left(x\right).$$

(20)

The solution of Equation (20) has the form

$${\psi }_{\mu }\left(x\right)={(-1)}^m{\ell }_{\mu }\left(x\right)\ast G_m\left(x\right)={(-1)}^m\sum _{\beta =0}^N\sum _{\nu =0}^t{(-1)}^{\nu }{G}_m^{\left(\nu \right)}(x-x_{\beta }).$$

(21)

The square of the norm of the function ${\psi }_{\mu }\left(x\right)$ in $L_2^{\left(m\right)}(0,1)$ coincides with the form $\varphi \left(C\right)$

$$\Vert {\psi }_{\mu }/L_2^{\left(m\right)}{\Vert }^2=({\ell }_{\mu }\left(x\right),{\psi }_{\mu }\left(x\right))=$$

$$=\sum _{\beta =0}^N\sum _{{\beta }^{\prime }=0}^N\sum _{\nu =0}^t\sum _{{\nu }^{\prime }=0}^t{(-1)}^{\nu +{\nu }^{\prime }}{C}_{\beta }^{\left(\nu \right)}{C}_{{\beta }^{\prime }}^{{\nu }^{\prime }}{G}_m^{\nu +{\nu }^{\prime }}(x_{\beta }-x_{{\beta }^{\prime }}).$$

It follows that for nonzero $C(C_{\beta }^{\left(\nu \right)},C_{{\beta }^{\prime }}^{\left({\nu }^{\prime }\right)})$, the function $\varphi \left(C\right)$ is strictly positive, i.e., the positivity of $\varphi \left(C\right)$ for such C follows from the positivity of the norm ${\psi }_{\mu }\left(x\right)$ in $L_2^{\left(m\right)}(0,1)$.

Lemma 1 is proved completely. □

Lemma 2.

If the matrix S of system (16) has a right inverse, then the matrix Q of the system (13) and (14) is non-degenerate.

Proof of Lemma 2. 

Let us write the homogeneous system corresponding to systems (13) and (14) in the following form

$$\sum _{\beta =0}^N\sum _{\nu =0}^t{\overline{C}}_{\beta }^{\left(\nu \right)}{G}_m^{(\nu +{\nu }^{\prime })}(x_{\beta }-x_{{\beta }^{\prime }})+\sum _{k={\nu }^{\prime }}^{m-1}{\overline{\lambda }}_k{\displaystyle \frac{k!}{(k-{\nu }^{\prime })!}}{x}_{\beta }^{(k-{\nu }^{\prime })}=0,$$

(22)

$${\beta }^{\prime }=\overline{0,N},{\nu }^{\prime }=\overline{0,t},$$

$$\sum _{\beta =0}^N\sum _{\nu =0}^t{\displaystyle \frac{k!}{(k-\nu )!}}{\overline{C}}_{\beta }^{\left(\nu \right)}{x}_{\beta }^{(k-\nu )}=0,k=\overline{0,m-1},$$

(23)

where

$$x_{\beta }^{(k-\nu )}=\left\{\begin{array}{cc}{x}_{\beta }^{(k-\nu )}\hfill & \mathrm{if}k-\nu \ge 0\hfill \\ 0\hfill & \mathrm{if}k-\nu <0.\hfill \end{array}\right.$$

Let us denote by G the matrix of quadratic form (18), and write the homogeneous systems (22) and (23) in the following form

$$G\left(\begin{array}{c}\overline{C}\\ \overline{\lambda }\end{array}\right)=\left(\begin{array}{cc}G& S^{*}\\ S& 0\end{array}\right)\left(\begin{array}{c}\overline{C}\\ \overline{\lambda }\end{array}\right)=0.$$

(24)

Now we prove that the only solution of the homogeneous system (24) is identically zero, i.e., $\overline{C}=0$ and $\overline{\lambda }=0$.

Let $\overline{C},\overline{\lambda }$ be the solution of system (24).

Consider the functional corresponding to the vector $\overline{C}$

$${\mu }_{\overline{C}}\left(x\right)=\sum _{\beta =0}^N\sum _{\nu =0}^t{(-1)}^{\nu }{\overline{C}}_{\beta }^{\left(\nu \right)}{\delta }^{\left(\nu \right)}(x-x_{\beta }).$$

Clearly, ${\mu }_{\overline{C}}\left(x\right)\in L_2^{\left(m\right)*}(0,1)$.

Let us take the following as the extremal function for the functional ${\mu }_{\overline{C}}\left(x\right)$:

$$U_{\overline{C}}\left(x\right)={(-1)}^m{\mu }_{\overline{C}}\left(x\right)\ast G_m\left(x\right)+\sum _{k={\nu }^{\prime }}^{m-1}{\overline{\lambda }}_k{\displaystyle \frac{k!}{(k-{\nu }^{\prime })!}}{x}^{(k-{\nu }^{\prime })}.$$

This is possible because $U_{\overline{C}}\left(x\right)\in L_2^{\left(m\right)}(0,1)$ and is a solution of Equation

$${\displaystyle \frac{d^{2m}{U}_{\overline{C}}\left(x\right)}{d{x}^{2m}}}={(-1)}^m{\mu }_{\overline{C}}\left(x\right).$$

The system of Equation (24) means that $U_{\overline{C}}^{\left(\nu \right)}\left(x\right)$ takes zero values at all nodes $x_{\beta }$, i.e., $U_{\overline{C}}^{\left(\nu \right)}\left(x_{\beta }\right)=0$, when $\beta =\overline{0,N},\nu =\overline{0,t}$. Then with respect to the norm in $L_2^{\left(m\right)*}(0,1)$ of the functional ${\mu }_{\overline{C}}\left(x\right)$, we have

$$\Vert {\mu }_{\overline{C}}/L_2^{\left(m\right)*}{\Vert }^2=({\mu }_{\overline{C}}\left(x\right),U_{\overline{C}}\left(x\right))=\sum _{\beta =0}^N\sum _{\nu =0}^t{C}_{\beta }^{\left(\nu \right)}{U}_{\overline{C}}^{\left(\nu \right)}\left(x_{\beta }\right)=0,$$

which is possible only at $\overline{C}=0$. Taking this into account from (24), we obtain

$$S^{*}\overline{\lambda }=0.$$

(25)

By convention, the matrix S has a right inverse, then $S^{*}$ has a left inverse. Hence and from (25) it follows that $\overline{\lambda }=0$.

Lemma 2 is proved completely. □

Thus, systems (13) and (14) have a single solution. Therefore, the vector $\stackrel{\circ }{C}$ delivers a local minimum to the quadratic function $F\left(C\right)$ on the set of solutions of system (5). The following theorem follows directly from Theorem 1 and Lemmas 1 and 2.

Theorem 2.

Let the error functional ${\ell }_N^{\left(t\right)}\left(x\right)$ of the quadrature Formula (3) be defined in the space $L_2^{\left(m\right)}(0,1)$, such that it vanishes on all polynomials of degree less than m and is optimal in the sense that, among all functionals of the form (4) with fixed nodes $x_{\beta }$, it has the minimal norm in $L_2^{\left(m\right)}(0,1)$. Under these conditions, there exists a function ${\psi }_{\ell }\left(x\right)$ that satisfies the corresponding equation

$${\displaystyle \frac{d^{2m}}{d{x}^{2m}}}{\psi }_{\ell }\left(x\right)={(-1)}^m{\ell }_N^{\left(t\right)}\left(x\right),$$

which goes to zero with its derivatives of order $\nu (\nu =\overline{0,t},t=\overline{0,m-1})$ at nodes $x_{\beta }$, i.e., ${\psi }_{\ell }^{\left(m\right)}\left(x_{\beta }\right)=0$ and belongs to $L_2^{\left(m\right)}(0,1)$.

Theorem 2 generalizes the theorem of I. Babushka [47], i.e., it is proved that the extremal function for the error functional goes to zero at nodes $x_{\beta }$.

3. An Approach to the Construction of Weighted Optimal Quadrature Formulas with Derivatives

In this section, we consider the following quadrature formula

$${\int }_0^{1}p\left(x\right)\phi \left(x\right)dx\cong \sum _{\beta =0}^N\sum _{\nu =0}^{m-1}{C}_{\beta }^{\left(\nu \right)}{\phi }^{\left(\nu \right)}\left(x_{\beta }\right),$$

(26)

i.e., the case $t=m-1$ in Formula (3). The error of this formula is

$$({\ell }_N^{(m-1)},\phi )={\int }_0^{1}p\left(x\right)\phi \left(x\right)dx-\sum _{\beta =0}^N\sum _{\nu =0}^{m-1}{C}_{\beta }^{\left(\nu \right)}{\phi }^{\left(\nu \right)}\left(x_{\beta }\right)=$$

$$={\int }_{-\infty }^{+\infty }{\ell }_N^{(m-1)}\left(x\right)\phi \left(x\right)dx,$$

where

$${\ell }_N^{(m-1)}\left(x\right)={\mathcal{E}}_{[0,1]}\left(x\right)p\left(x\right)-\sum _{\nu =0}^{m-1}\sum _{\beta =0}^N{(-1)}^{\nu }{C}_{\beta }^{\left(\nu \right)}{\delta }^{\left(\nu \right)}(x-x_{\beta }).$$

(27)

The quadrature Formula (26) with error functional (27), considered in the space $L_2^{\left(m\right)}(0,1)$, can be characterized in two ways. On the one hand, it is defined by the coefficients $C_{\beta }^{\left(\nu \right)}(\nu =\overline{0,m-1},\beta =\overline{0,N})$ subject to the conditions:

$$({\ell }_N^{(m-1)}\left(x\right),x^{\alpha })=0,\alpha =\overline{0,m-1},$$

(28)

and, on the other hand, the extreme function ${\psi }_{\ell }\left(x\right)$ of the quadrature formula, which is obtained as a solution of Equation

$${\displaystyle \frac{d^{2m}U\left(x\right)}{d{x}^{2m}}}={(-1)}^m{\ell }_N^{(m-1)}\left(x\right)$$

and can be written in the form

$${\psi }_{\ell }\left(x\right)={(-1)}^m{\ell }_N^{(m-1)}\left(x\right)\ast G_m\left(x\right)+P_{m-1}\left(x\right),$$

(29)

where

$$G_m\left(x\right)={\displaystyle \frac{x^{2m-1}sign\left(x\right)}{2(2m-1)!}},$$

$P_{m-1}\left(x\right)$ is some polynomial of degree $m-1$.

In this case, the square of the norm of the error functional ${\ell }_N^{(m-1)}\left(x\right)$ is calculated by the formula

$$\Vert {\ell }_N^{(m-1)}{\Vert }^2=({\ell }_N^{(m-1)},{\psi }_{\ell }).$$

(30)

As previously established, the function ${\psi }_{\ell }\left(x\right)$ is given by formula (29). Therefore, by computing the square of the norm of the error functional with the help of Equation (30), we arrive at

$$\Vert {\ell }_N^{(m-1)}{\Vert }^2={(-1)}^m[\sum _{\beta =0}^N\sum _{{\beta }^{\prime }=0}^N\sum _{\nu =0}^{m-1}\sum _{{\nu }^{\prime }=0}^{m-1}{(-1)}^{\nu +{\nu }^{\prime }}{C}_{\beta }^{\left(\nu \right)}{C}_{{\beta }^{\prime }}^{{\nu }^{\prime }}{G}_m^{(\nu +{\nu }^{\prime })}(x_{\beta }-x_{{\beta }^{\prime }})-$$

$$-2\sum _{\nu =0}^{m-1}\sum _{\beta =0}^N{(-1)}^{\nu }{C}_{\beta }^{\left(\nu \right)}{\int }_0^{1}p\left(x\right)G_m^{\left(\nu \right)}(x-x_{\beta })dx+$$

$$+{\int }_0^1{\int }_0^{1}p\left(x\right)p\left(y\right)G_m(x-y)dxdy,$$

(31)

where

$$G_m^{\left(k\right)}\left(x\right)={\displaystyle \frac{x^{2m-1-k}}{2(2m-1-k)!}},k\le 2m-1.$$

4. Method for Constructing Weighted Optimal Quadrature Formulas Incorporating Derivatives

Let $x_{\beta }=\left[\beta \right],h={\displaystyle \frac{1}{N}},N=\overline{1,N},N\ge m,C_{\beta }^{\left(\nu \right)}=C^{\left(\nu \right)}\left[\beta \right],\left[\beta \right]=h\beta$.

Our approach to constructing optimal quadrature formulas with derivatives proceeds as follows. Initially, for $m=1$, that is, within the space $L_2^{\left(1\right)}(0,1)$, by minimizing the squared norm of the error functional (31) concerning the coefficients $C_{\beta }^{\left(0\right)}$ (for $\beta =\overline{0,N}$) under the constraints given by (28), we derive the following system for determining $C_{\beta }^{\left(0\right)}$:

$$\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\gamma }^{\left(0\right)}{\displaystyle \frac{(h\beta -h\gamma )sign(h\beta -h\gamma )}{2}}+{\lambda }_0=$$

$$={\int }_0^{1}p\left(x\right){\displaystyle \frac{(x-h\beta )sign(x-h\beta )}{2}}dx,\beta =\overline{0,N},$$

(32)

$$\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\gamma }^{\left(0\right)}={\int }_0^{1}p\left(x\right)dx.$$

(33)

Here (33) is obtained from (28) when $m=1$. Systems (32) and (33) are solved in [8], i.e., here we find the optimal coefficients ${\stackrel{\circ }{C}}_{\gamma }^{\left(0\right)}$ in the space $L_2^{\left(1\right)}(0,1)$. The application of this quadrature formula to the linear integral equations given in [48].

Next, let us consider the case m = 2. For this purpose, by substituting the found optimal coefficients into (31), then minimizing the square of the norm on the coefficients ${\stackrel{\circ }{C}}_{\gamma }^{\left(1\right)}$ in the space $L_2^{\left(2\right)}(0,1)$, we obtain the optimal coefficients ${\stackrel{\circ }{C}}_{\gamma }^{\left(1\right)}(\beta =\overline{0,N})$.

Continuing this method, we successively find the optimal coefficients ${\stackrel{\circ }{C}}_{\gamma }^{\left(0\right)},{\stackrel{\circ }{C}}_{\gamma }^{\left(1\right)},\cdots ,{\stackrel{\circ }{C}}_{\gamma }^{(k-1)}$. Substituting these coefficients into (31) and minimizing the square of the norm on the coefficients of ${\stackrel{\circ }{C}}_{\gamma }^{(k-1)}$ in the space $L_2^{\left(k\right)}(0,1)$, we obtain a system for finding the optimal coefficients of ${\stackrel{\circ }{C}}_{\gamma }^{(k-1)}(\beta =\overline{0,N})$:

$$\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\gamma }^{\left(k\right)}{\displaystyle \frac{(h\beta -h\gamma )sign(h\beta -h\gamma )}{2}}+{(-1)}^{k}k!{\lambda }_k={(-1)}^k{F}_{k\beta },$$

(34)

$$\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\beta }^{\left(k\right)}={\displaystyle \frac{g_k}{k!}}-\sum _{i=0}^{k-1}\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\gamma }^{\left(i\right)}{\displaystyle \frac{{\left(h\gamma \right)}^{k-i}}{(k-i)!}},\beta =\overline{0,N},$$

(35)

Here

$$F_{k\beta }=f_{k\beta }-\sum _{i=0}^{k-1}\sum _{\gamma =0}^N{(-1)}^i{\stackrel{\circ }{C}}_{\gamma }^{\left(i\right)}{\displaystyle \frac{{(h\beta -h\gamma )}^{k-i+1}sign(h\beta -h\gamma )}{2(k-i+1)!}},$$

(36)

$$f_{k\beta }={(-1)}^k{\int }_0^{1}p\left(x\right){\displaystyle \frac{{(x-h\beta )}^{k+1}sign(x-h\beta )}{2(k+1)!}}dx,$$

$$g_k={\int }_0^{1}p\left(x\right)x^{k}dx,k=\overline{0,m-1}.$$

5. Optimal Coefficients of Weighted Quadrature Formulas with Derivative

We now solve the system of linear algebraic Equations (34) and (35) with respect to ${\stackrel{\circ }{C}}_{\beta }^{\left(k\right)}(\beta =\overline{0,N},k=\overline{0,m-1})$, the optimal coefficients of weight quadrature formulas with derivatives.

Let us rewrite systems (34) and (35) in the following form

$$\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\gamma }^{\left(k\right)}{G}_1\left[\beta \right](h\beta -h\gamma )+{(-1)}^{k}k!{\lambda }_k={(-1)}^k{F}_{k\beta },$$

(37)

$$\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\gamma }^{\left(k\right)}=p_k,\beta =\overline{0,N},k=\overline{0,m-1}.$$

(38)

Here ${\lambda }_k$ is a constant, and $F_{k\beta }$ is determined by the equality (36),

$$p_k={\displaystyle \frac{g_k}{k!}}-\sum _{i=0}^{k-1}\sum _{\gamma =0}^N{\stackrel{\circ }{C}}_{\gamma }^{\left(i\right)}{\left(h\gamma \right)}^{k-i}(k-i)!,G_1\left(h\beta \right)={\displaystyle \frac{\left|h\beta \right|}{2}},k=\overline{0,m-1}.$$

The coefficients of the first term of the first equation depend only on the difference $(h\beta -h\gamma )$. These kinds of equations in the continuous case, where instead of the sum there are integrals, are called Wiener–Hopf equations. As is typical for Wiener–Hopf equations, we assume that ${\stackrel{\circ }{C}}_{\gamma }^{\left(k\right)}$ is defined everywhere, i.e., $\gamma \in Z$, and equal to zero if $h\gamma \notin [0,1]$.

Let us further assume ${\stackrel{\circ }{C}}_{\beta }^{\left(k\right)}={\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right],G_1\left[\beta \right]=G_1\left(h\beta \right),F_{k\beta }=F_k\left[\beta \right]$. Then systems (37) and (38) are written in the form of convolution equations

$$G_1\left[\beta \right]\ast {\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]+{(-1)}^{k}k!{\lambda }_k={(-1)}^k{F}_k\left[\beta \right],\beta =\overline{0,N},$$

(39)

$$\sum _{\gamma =0}^N{\stackrel{\circ }{C}}^{\left(k\right)}\left[\gamma \right]=p_k,$$

(40)

$${\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]=0,\left[\beta \right]\notin [0,1].$$

(41)

We now proceed to the actual solution of systems (39) and (41). For this purpose, instead of ${\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]$ we introduce the following function

$$u\left[\beta \right]=G_1\left[\beta \right]\ast {\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]+{(-1)}^{k}k!{\lambda }_k.$$

Next, we define the function $u\left[\beta \right]$ when $\beta \le 0$ and $\beta \ge N$.

Let $\beta \le 0$, then by virtue of (40) we obtain

$$u\left[\beta \right]=-{\displaystyle \frac{h\beta p_k}{2}}+a_k^{-},$$

$a_k^{-}$ is unknown.

Let $\beta \ge N$, then

$$u\left[\beta \right]={\displaystyle \frac{h\beta p_k}{2}}+a_k^{+},$$

$a_k^{+}$ is unknown.

So, we have defined a function $u\left[\beta \right]$ for all values of $\left[\beta \right]\in Z$:

$$u\left[\beta \right]=\left\{\begin{array}{cc}-{\displaystyle \frac{h\beta p_k}{2}}+a_k^{-}\hfill & \mathrm{if}\beta \le 0,\hfill \\ {(-1)}^k{F}_k\left[\beta \right]\hfill & \mathrm{if}0\le \beta \le N,\hfill \\ {\displaystyle \frac{h\beta p_k}{2}}+a_k^{+}\hfill & \mathrm{if}\beta \ge N.\hfill \end{array}\right.$$

(42)

Since in (42) at $\beta =0$ and $\beta =N$ the left and right parts coincide, we have

$$a_k^{-}={(-1)}^k{F}_k\left[0\right],a_k^{+}={(-1)}^k{F}_k\left[N\right]-{\displaystyle \frac{p_k}{2}}.$$

Then we obtain a new representation of the function $u\left[\beta \right]$ in $\left[\beta \right]\in Z$:

$$u\left[\beta \right]=\left\{\begin{array}{cc}-{\displaystyle \frac{h\beta p_k}{2}}+{(-1)}^k{F}_k\left[0\right]\hfill & \mathrm{if}\beta \le 0,\hfill \\ {(-1)}^k{F}_k\left[\beta \right]\hfill & \mathrm{if}0\le \beta \le N,\hfill \\ {\displaystyle \frac{h\beta p_k}{2}}+{(-1)}^k{F}_k\left[N\right]-{\displaystyle \frac{p_k}{2}}\hfill & \mathrm{if}\beta \ge N.\hfill \end{array}\right.$$

(43)

Now we will need the well-known formula [9]:

$$h{D}_1\left[\beta \right]\ast G_1\left[\beta \right]=\delta \left[\beta \right],$$

(44)

where

$$D_1\left[\beta \right]=\left\{\begin{array}{cc}0\hfill & \mathrm{if}\left[\beta \right]\ge 2,\hfill \\ h^{-2}\hfill & \mathrm{if}\left[\beta \right]=1,\hfill \\ -2h^{-2}\hfill & \mathrm{if}\left[\beta \right]=0,\hfill \end{array}\right.$$

(45)

$$\delta \left[\beta \right]=\left\{\begin{array}{cc}0\hfill & \mathrm{at}\left[\beta \right]\ne 0,\hfill \\ 1\hfill & \mathrm{at}\left[\beta \right]=0.\hfill \end{array}\right.$$

(46)

By virtue of Formulas (43) and (44), we find the optimal coefficients ${\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]$ at $\beta =\overline{0,N}$:

$${\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]=h{D}_1\left[\beta \right]\ast u\left[\beta \right]=h\sum _{\gamma =-\infty }^{\infty }{D}_1[\beta -\gamma ]u\left[\gamma \right]=$$

$$h[{(-1)}^k\sum _{\gamma =0}^N{D}_1[\beta -\gamma ]F_k\left[\gamma \right]+$$

$$+\sum _{\gamma =-\infty }^{-1}{D}_1[\beta -\gamma ](-{\displaystyle \frac{h\gamma p_k}{2}}+{(-1)}^k{F}_k\left[0\right])+$$

$$+\sum _{\gamma =N+1}^{\infty }{D}_1[\beta -\gamma ]({\displaystyle \frac{h\gamma p_k}{2}}+{(-1)}^k{F}_k\left[N\right]-{\displaystyle \frac{p_k}{2}})],k=\overline{0,m-1}.$$

Hence, using Formula (45), we obtain

$${\stackrel{\circ }{C}}^k\left[0\right]={\displaystyle \frac{p_k}{2}}+{(-1)}^k{h}^{-1}[F_k\left[1\right]-F_k\left[0\right]],$$

(47)

$${\stackrel{\circ }{C}}^k\left[\beta \right]={(-1)}^k{h}^{-1}[F_k[\beta -1]-2F_k\left[\beta \right]+F_k[beta+1]],\beta =\overline{0,N},$$

(48)

$${\stackrel{\circ }{C}}^k\left[N\right]={\displaystyle \frac{p_k}{2}}+{(-1)}^k{h}^{-1}[F_k[N-1]-F_k\left[N\right]],k=\overline{0,m-1}$$

(49)

Thus, the following theorem is proved.

Theorem 3.

The optimal coefficients of the quadrature formula of the form (26) in the Sobolev space $L_2^{\left(m\right)}(0,1)$ are determined by Formulas (47)–(49).

From Theorem 3, when $p\left(x\right)\equiv 1$ we obtain:

Corollary 1.

The optimal coefficients of the quadrature formula of the form (26) at $p\left(x\right)\equiv 1$ in the Sobolev space $L_2^{\left(m\right)}(0,1)$ are defined by the formulas

$${\stackrel{\circ }{C}}^{\left(0\right)}\left[0\right]={\displaystyle \frac{h}{2}},{\stackrel{\circ }{C}}^{\left(0\right)}\left[\beta \right]=h,{\stackrel{\circ }{C}}^{\left(0\right)}\left[N\right]={\displaystyle \frac{h}{2}},$$

$${\stackrel{\circ }{C}}^{\left(k\right)}\left[0\right]={\displaystyle \frac{{(-1)}^k{h}^{k+1}{B}_{k+1}}{(k+1)!}},$$

$${\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]=0,\beta =\overline{1,N-1},$$

$${\stackrel{\circ }{C}}^{\left(k\right)}\left[N\right]={\displaystyle \frac{{(-1)}^k{h}^{k+1}{B}_{k+1}}{(k+1)!}},k=\overline{1,m-1},$$

where $B_{k+1}$ are the Bernoulli numbers.

Corollary 1 demonstrates that the optimal coefficients coincide with those of the classical Euler–Maclaurin quadrature formula. The optimality of the Euler–Maclaurin quadrature formula in the space $L_2^{\left(m\right)}(0,1)$ has been established in [8,9].

For completeness, we give the following theorem.

Theorem 4.

Euler–Maclorean quadrature formula

$${\int }_0^1\phi \left(x\right)dx\cong h\sum _{n=1}^N\phi \left(nh\right)+\sum _{\alpha =0}^{m-1}{(-1)}^{\alpha }{h}^{\alpha +1}{\displaystyle \frac{B_{\alpha +1}}{(\alpha +1)!}}({\phi }^{\left(\alpha \right)}\left(1\right)-{\phi }^{\left(\alpha \right)}\left(0\right))$$

with the error functional

$$\ell \left(x\right)={\mathcal{E}}_{[0,1]}\left(x\right)-h\sum _{n=1}^N\delta (x-hn)-\sum _{\alpha =0}^{m-1}{\displaystyle \frac{h^{\alpha +1}{B}_{\alpha +1}}{(\alpha +1)!}}({\delta }^{\left(\alpha \right)}(x-1)-{\delta }^{\left(\alpha \right)}\left(x\right))$$

is the optimal quadrature formula in Sobolev space $L_2^{\left(m\right)}(0,1)$. The square of the norm of the error functional of the optimal Euler–Maclorean quadrature formula is defined by the following equality

$$\Vert \stackrel{\circ }{\ell }{|L_2^{\left(m\right)*}(0,1)\Vert }^2={\left({\displaystyle \frac{h}{2\pi }}\right)}^{2m}\sum _{\gamma \ne 0}{\displaystyle \frac{1}{{\gamma }^{2m}}}.$$

Here $h={\displaystyle \frac{1}{N}},N=2,3,\dots ,B_{\alpha +1}$ are Bernoulli numbers, $B_1=-{\displaystyle \frac{1}{2}}$,

$$\delta \left[\beta \right]=\left\{\begin{array}{cc}0\hfill & \mathrm{at}\left[\beta \right]\ne 0,\hfill \\ 1\hfill & \mathrm{at}\left[\beta \right]=0.\hfill \end{array}\right.$$

$${\displaystyle \frac{B_{\alpha +1}}{(\alpha +1)!}}=\left\{\begin{array}{cc}-{(-1)}^{{\displaystyle \frac{\alpha +1}{2}}}\sum _{\gamma \ne 0}{\displaystyle \frac{1}{{\left(2\pi \gamma \right)}^{\alpha +1}}}\hfill & \mathrm{if}\alpha +1-\mathit{even}\mathit{number},\hfill \\ 0\hfill & \mathrm{if}\alpha +1-\mathit{odd}\mathit{number}.\hfill \end{array}\right.$$

From Theorem 3, after some simplifications, we obtain the following results in the spaces $L_2^{\left(1\right)}(0,1),L_2^{\left(2\right)}(0,1),L_2^{\left(3\right)}(0,1)$.

Theorem 5.

In the Sobolev space $L_2^{\left(1\right)}(0,1)$, the following quadrature formula is the optimal quadrature formula

$${\int }_0^{1}p\left(x\right)\phi \left(x\right)dx\cong \sum _{\beta =0}^N{\stackrel{\circ }{C}}^{\left(0\right)}\left[\beta \right]\phi \left[\beta \right],$$

where

$${\stackrel{\circ }{C}}^{\left(0\right)}\left[0\right]=h^{-1}{\int }_0^{h}p\left(x\right)(h-x)dx,$$

$${\stackrel{\circ }{C}}^{\left(0\right)}\left[\beta \right]=h^{-1}{\int }_{h(\beta -1)}^{h(\beta +1)}p\left(x\right)[(h\beta -x)sign(x-h\beta )+h]dx,\beta =\overline{1,N-1},$$

$${\stackrel{\circ }{C}}^{\left(0\right)}\left[N\right]=h^{-1}{\int }_{1-h}^{1}p\left(x\right)(x-1+h)dx.$$

Theorem 6.

In the Sobolev space $L_2^{\left(2\right)}(0,1)$, the optimal quadrature formula is the following formula

$${\int }_0^{1}p\left(x\right)\phi \left(x\right)dx\cong \sum _{\beta =0}^N\left({\stackrel{\circ }{C}}^{\left(0\right)}\left[\beta \right]\phi \left[\beta \right]+{\stackrel{\circ }{C}}^{\left(1\right)}\left[\beta \right]{\phi }^{\prime }\left[\beta \right]\right).$$

Here ${\stackrel{\circ }{C}}^{\left(0\right)}\left[\beta \right]$ are determined from Theorem 5, and the optimal coefficients ${\stackrel{\circ }{C}}^{\left(1\right)}\left[\beta \right]$ are computed by the following formulas below:

$${\stackrel{\circ }{C}}^{\left(1\right)}\left[0\right]=-{\displaystyle \frac{h^{-1}}{2}}{\int }_0^{h}p\left(x\right)(x-h)xdx,$$

$${\stackrel{\circ }{C}}^{\left(1\right)}\left[\beta \right]={\displaystyle \frac{h^{-1}}{2}}{\int }_{h(\beta -1)}^{h(\beta +1)}p\left(x\right)[{(x-h\beta )}^{2}sign(h\beta -x)+h(x-h\beta )]dx,\beta =\overline{1,N-1},$$

$${\stackrel{\circ }{C}}^{\left(1\right)}\left[N\right]={\displaystyle \frac{h^{-1}}{2}}{\int }_{1-h}^{1}p\left(x\right)(x-1+h)(x-1)dx.$$

Theorem 7.

The optimal coefficients ${\stackrel{\circ }{C}}^{\left(2\right)}\left[\beta \right]$ of the quadrature formula of the form

$${\int }_0^{1}p\left(x\right)\phi \left(x\right)dx\cong \sum _{\beta =0}^N\left({\stackrel{\circ }{C}}^{\left(0\right)}\left[\beta \right]\phi \left[\beta \right]+{\stackrel{\circ }{C}}^{\left(1\right)}\left[\beta \right]{\phi }^{\prime }\left[\beta \right]+{\stackrel{\circ }{C}}^{\left(2\right)}\left[\beta \right]{\phi }^{″}\left[\beta \right]\right),$$

in the space $L_2^{\left(3\right)}(0,1)$ are defined by the formulas

$${\stackrel{\circ }{C}}^{\left(2\right)}\left[0\right]={\displaystyle \frac{h^{-1}}{12}}{\int }_0^{h}p\left(x\right)x(h-x)(2x-h)xdx,$$

$$\beta =\overline{1,N-1},$$

$${\stackrel{\circ }{C}}^{\left(2\right)}\left[\beta \right]={\displaystyle \frac{h^{-1}}{12}}{\int }_{h(\beta -1)}^{h(\beta +1)}p\left(x\right)[(2{(x-h\beta )}^3+h^2(x-h\beta ))sign(h\beta -x)+3h{(x-h\beta )}^2]dx,$$

$${\stackrel{\circ }{C}}^{\left(2\right)}\left[N\right]={\displaystyle \frac{h^{-1}}{12}}{\int }_{1-h}^{1}p\left(x\right)(x-1)(2x+h-2)(x+h-1)dx.$$

6. 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

$$y\left(x\right)-\lambda {\int }_0^{1}K(x,s)y\left(s\right)ds=f\left(x\right),x\in [0,1],$$

(50)

where $K(x,s)$ is the kernel of the integral equation, $f\left(x\right)$ is the right-hand side, $\lambda$ is the parameter of the integral equation and $y\left(x\right)$ 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, [49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67]).

To solve Equation (50) we apply the quadrature Formula (26) and pass the difference grid on the argument x

$$y_i-\lambda \sum _{k=0}^{m-1}\sum _{\beta =0}^N{C}_{i\beta }^{\left(k\right)}{y}_{\beta }^{\left(k\right)}=f_i,i=\overline{0,N},m=1,2,\cdots ,$$

(51)

Here $C_{\beta }^{\left(k\right)}$ are the optimal coefficients of the quadrature formula, $f_i=f\left(x_i\right),y_i=y\left(x_i\right),y_i^{\left(k\right)}=y^{\left(k\right)}\left(x_i\right),x_i=i*h,i=\overline{0,N},h$ 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

$$y\left(x\right)-\lambda {\int }_0^{1}K(x,s)y\left(s\right)ds=f\left(x\right),x\in [0,1],$$

$$y^{\prime }\left(x\right)-\lambda {\int }_0^1{K}_x^{\prime }(x,s)y\left(s\right)ds=f^{\prime }\left(x\right),$$

(52)

$$\cdots$$

$$y^{(m-1)}\left(x\right)-\lambda {\int }_0^1{K}_x^{(m-1)}(x,s)y\left(s\right)ds=f^{(m-1)}\left(x\right),m=1,2,\cdots .$$

Now applying the quadrature formula to system (52), we obtain

$$y_i-\lambda \sum _{k=0}^{m-1}\sum _{\beta =0}^N{C}_{i\beta }^{\left(k\right)}{y}_{\beta }^{\left(k\right)}=f_i,i=\overline{0,N},$$

$$y_i^{\prime }-\lambda \sum _{k=0}^{m-1}\sum _{\beta =0}^N{C}_{i\beta }^{\left(k\right)}{y}_{\beta }^{\left(k\right)}=f_i^{\prime },$$

(53)

$$\cdots$$

$$y_i^{(m-1)}-\lambda \sum _{k=0}^{m-1}\sum _{\beta =0}^N{C}_{i\beta }^{\left(k\right)}{y}_{\beta }^{\left(k\right)}=f_i^{(m-1)},m=1,2,\cdots .$$

Thus, we have a system of linear algebraic equations with respect $y_i^{\left(k\right)}(i=\overline{0,N},k=\overline{0,m-1})$. The values of the desired function $y_i=y_i^{\left(0\right)}(i=\overline{0,N})$.

7. Numerical Results

The following examples illustrate the solution of integral equations using the quadrature method based on the optimal quadrature formula with derivatives for $m=2,4,8$. The obtained results are compared with the exact solutions and with those reported by other researchers, using the absolute error as the measure of accuracy.

$$E_m=\underset{x}{\max }|y\left(x\right)-y_{ex}\left(x\right)|.$$

Here $E_m$ is the maximum absolute error, $y\left(x\right)$ is the approximate solution and $y_{ex}\left(x\right)$ is the exact solution.

It should be noted that a program was created with the Maple language for the above algorithm. All calculations are performed using 32 significant digits.

Example 1.

In (50) $K(x,s)=-x\times (e^{x\times s}-1),f\left(x\right)=e^x-x,\lambda =1,a=0,b=1$. Then integral Equation (61) will take the following form:

$$y\left(x\right)+{\int }_0^{1}x\times (e^{x\times s}-1)\times y\left(s\right)ds=e^x-x,x\in [0,1]$$

(54)

The exact solution of integral Equation (54) is as follows:

$$y_{ex}\left(x\right)=1.$$

The results for this example are summarized in

Table 1. Results of solving the integral equation in Example 1.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$1.000$	$1.000$	$1.000$	$1.000$	$0.00E+00$	$0.00E+00$	$0.00E+00$
0.1	$1.000$	$1.000$	$1.000$	$1.000$	$1.60E-31$	$0.00E+00$	$1.80E-31$
0.2	$1.000$	$1.000$	$1.000$	$1.000$	$4.00E-32$	$8.00E-32$	$4.00E-31$
0.3	$1.000$	$1.000$	$1.000$	$1.000$	$1.70E-31$	$2.10E-31$	$1.40E-31$
0.4	$1.000$	$1.000$	$1.000$	$1.000$	$1.00E-31$	$1.00E-31$	$1.00E-31$
0.5	$1.000$	$1.000$	$1.000$	$1.000$	$2.00E-32$	$0.00E+00$	$5.00E-32$
0.6	$1.000$	$1.000$	$1.000$	$1.000$	$1.00E-31$	$2.00E-31$	$2.80E-31$
0.7	$1.000$	$1.000$	$1.000$	$1.000$	$0.00E+00$	$1.00E-31$	$3.00E-31$
0.8	$1.000$	$1.000$	$1.000$	$1.000$	$5.00E-32$	$1.60E-31$	$2.00E-31$
0.9	$1.000$	$1.000$	$1.000$	$1.000$	$1.00E-31$	$1.00E-31$	$1.00E-31$
1.0	$1.000$	$1.000$	$1.000$	$1.000$	$1.00E-31$	$1.00E-31$	$1.00E-31$
				MAE	$1.70E-31$	$3.00E-31$	$4.00E-31$

.

Here ES is the exact solution, OQF is the optimal quadrature formula and MAE is the maximum absolute error.

The authors of [29] solved this example using a modified multistage average integral method and obtained a result with the maximum absolute error $E_m=3.44\times {10}^{-15}$ for $N=13$ equidistant collocation nodes.

On the basis of using the method of integral replacement with the twelfth-order quadrature formula, the authors of [45] obtained the result with the maximum absolute error $E_m=4.44\times {10}^{-16}$ at $N=20$ integration intervals.

Our method gave the result with the value of the maximum absolute error $E_m=4.0\times {10}^{-31}$ at $m=1$.

Example 2.

In (50), $K(x,s)=x\times s,f\left(x\right)=\cos \left(x\right),\lambda =1,a=0,b=1$. Then integral Equation (50) takes the following form:

$$y\left(x\right)-{\int }_0^{1}x\times s\times y\left(s\right)ds=\cos \left(x\right),x\in [0,1].$$

(55)

The exact solution of integral Equation (55) is as follows:

$$y_{ex}\left(x\right)=cos\left(x\right)+1.5\times (\sin \left(1\right)+\cos \left(1\right)-1)\times x.$$

The results for this example are shown in

Table 2. Results of solving the integral equation in Example 2.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$1.000$	$1.000$	$1.000$	$1.000$	$0.00E+00$	$0.00E+00$	$0.00E+00$
0.1	$1.052$	$1.052$	$1.052$	$1.052$	$1.62E-09$	$3.85E-13$	$2.43E-20$
0.2	$1.095$	$1.095$	$1.095$	$1.095$	$3.23E-09$	$7.69E-13$	$4.86E-20$
0.3	$1.127$	$1.127$	$1.127$	$1.127$	$4.85E-09$	$1.15E-12$	$7.29E-20$
0.4	$1.150$	$1.150$	$1.150$	$1.150$	$6.46E-09$	$1.54E-12$	$9.72E-20$
0.5	$1.164$	$1.164$	$1.164$	$1.164$	$8.08E-09$	$1.92E-12$	$1.21E-19$
0.6	$1.169$	$1.169$	$1.169$	$1.169$	$9.69E-09$	$2.31E-12$	$1.46E-19$
0.7	$1.166$	$1.166$	$1.166$	$1.166$	$1.13E-08$	$2.69E-12$	$1.70E-19$
0.8	$1.155$	$1.155$	$1.155$	$1.155$	$1.29E-08$	$3.08E-12$	$1.94E-19$
0.9	$1.137$	$1.137$	$1.137$	$1.137$	$1.45E-08$	$3.46E-12$	$2.19E-19$
1.0	$1.113$	$1.113$	$1.113$	$1.113$	$1.62E-08$	$3.85E-12$	$2.43E-19$
				MAE	$1.62E-08$	$3.85E-12$	$2.43E-19$

.

To solve the Fredholm integral equation of the second kind, the authors of [30] developed a method using spline technology. Using this method, they obtained the result for this example. The maximum absolute error was $E_m=2.0\times {10}^{-5}$ at $N=11,n=10$. Our method gave the result with the value of maximum absolute error $E_m=1.62\times {10}^{-8}-2.43\times {10}^{-19}$ at $m=\overline{3,8}$.

Example 3.

In (50), $K(x,s)=x^2-x-s^2+s,f\left(x\right)=-2x^3+3x^2-x,\lambda =-1,a=0,b=1$. Then integral Equation (50) will take the following form:

$$y\left(x\right)+{\int }_0^1(x^2-x-s^2+s)y\left(s\right)ds=-2x^3+3x^2-x,x\in [0,1].$$

(56)

The exact solution of integral Equation (56) is as follows:

$$y_{ex}\left(x\right)=-2x^3+3x^2-x.$$

The results for this example are given in

Table 3. Results of solving the integral equation in Example 3.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$0.000$	$0.000$	$0.000$	$0.000$	$4.30E-33$	$2.47E-32$	$2.47E-32$
0.1	$-0.072$	$-0.072$	$-0.072$	$-0.072$	$1.10E-32$	$5.00E-33$	$5.00E-33$
0.2	$-0.096$	$-0.096$	$-0.096$	$-0.096$	$7.00E-33$	$4.00E-33$	$4.00E-33$
0.3	$-0.084$	$-0.084$	$-0.084$	$-0.084$	$6.00E-33$	$1.20E-32$	$1.20E-32$
0.4	$-0.048$	$-0.048$	$-0.048$	$-0.048$	$1.00E-33$	$3.00E-33$	$3.00E-33$
0.5	$0.000$	$0.000$	$0.000$	$0.000$	$7.80E-34$	$1.46E-32$	$1.46E-32$
0.6	$0.048$	$0.048$	$0.048$	$0.048$	$2.00E-33$	$3.00E-33$	$3.00E-33$
0.7	$0.084$	$0.084$	$0.084$	$0.084$	$1.70E-32$	$4.00E-33$	$4.00E-33$
0.8	$0.096$	$0.096$	$0.096$	$0.096$	$1.00E-32$	$1.30E-32$	$1.30E-32$
0.9	$0.072$	$0.072$	$0.072$	$0.072$	$1.00E-33$	$8.00E-33$	$8.00E-33$
1.0	$0.000$	$0.000$	$0.000$	$0.000$	$4.38E-33$	$2.47E-32$	$2.47E-32$
				MAE	$1.70E-32$	$2.47E-32$	$2.47E-32$

.

The authors of [42] developed a graph-theoretic polynomial using Hosoi polynomials and solved integral Equation (56). They obtained the result with the maximum absolute error $E_m=8.88\times {10}^{-16}$ at $n=3$.

The authors of [43] solved integral Equation (56) and obtained the result with the maximum absolute error $E_m=4.97\times {10}^{-4}$ at $N=8$.

Our method gave the result with the value of the maximum absolute error $E_m=1.0\times {10}^{-32}$ at $m=1$.

Example 4.

In (50) $K(x,s)=x\times e^s,f\left(x\right)=e^{-x},\lambda =-1,a=0,b=1$. Then integral Equation (50) will take the following form:

$$y\left(x\right)+{\int }_0^{1}x\times e^s\times y\left(s\right)ds=e^{-x},x\in [0,1].$$

(57)

The exact solution of integral Equation (57) is as follows:

$$y_{ex}\left(x\right)=e^{-x}-{\displaystyle \frac{x}{2}}.$$

The results for this example are given in

Table 4. Results of solving the integral equation in Example 4.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$1.000$	$1.000$	$1.000$	$1.000$	$0.00E+00$	$0.00E+00$	$0.00E+00$
0.0	$1.000$	$1.000$	$1.000$	$1.000$	$0.00E+00$	$0.00E+00$	$0.00E+00$
0.1	$0.855$	$0.855$	$0.855$	$0.855$	$1.39E-08$	$3.31E-12$	$2.09E-19$
0.2	$0.719$	$0.719$	$0.719$	$0.719$	$2.78E-08$	$6.62E-12$	$4.18E-19$
0.3	$0.591$	$0.591$	$0.591$	$0.591$	$4.17E-08$	$9.92E-12$	$6.27E-19$
0.4	$0.470$	$0.470$	$0.470$	$0.470$	$5.56E-08$	$1.32E-11$	$8.35E-19$
0.5	$0.357$	$0.357$	$0.357$	$0.357$	$6.95E-08$	$1.65E-11$	$1.04E-18$
0.6	$0.249$	$0.249$	$0.249$	$0.249$	$8.34E-08$	$1.98E-11$	$1.25E-18$
0.7	$0.147$	$0.147$	$0.147$	$0.147$	$9.73E-08$	$2.32E-11$	$1.46E-18$
0.8	$0.049$	$0.049$	$0.049$	$0.049$	$1.11E-07$	$2.65E-11$	$1.67E-18$
0.9	$-0.043$	$-0.043$	$-0.043$	$-0.043$	$1.25E-07$	$2.98E-11$	$1.88E-18$
1.0	$-0.132$	$-0.132$	$-0.132$	$-0.132$	$1.39E-07$	$3.31E-11$	$2.09E-18$
				MAE	$1.39E-07$	$3.31E-11$	$2.09E-18$

.

To solve integral Equation (57), the author of [32] applied the polynomial method using Boubaker polynomials and obtained the result with the maximum absolute error $E_m=7.1\times {10}^{-3}$ at $N=11$.

The authors of [36] also applied the polynomial method. They used the Tushar polynomial and obtained the result with the maximum absolute error $E_m=2.0\times {10}^{-3}$ at $N=11$ for $n=2$.

Using the Gauss–Lobatto quadrature formula, the author of [41] obtained good results. The maximum absolute error was $E_m=1.28\times {10}^{-15}$ at $N=11$ and $n=6$.

The authors of [44] applied the polynomial method using Bernstein polynomials to solve integral Equation (57) and obtained the result with the maximum absolute error $E_m=3.5\times {10}^{-3}$ at $N=11$ and $n=6$.

Our method gave the result with the value of maximum absolute error $E_m=2.09\times {10}^{-18}$ at $m=8$.

Example 5.

In (50), $K(x,s)=e^{x+s},f\left(x\right)=e^{-x},\lambda =-1,a=0,b=1$. Then the integral Equation (50) will take the following form:

$$y\left(x\right)+{\int }_0^1{e}^{x+s}\times y\left(s\right)ds=e^{-x},x\in [0,1]$$

(58)

The exact solution of integral Equation (58) is as follows:

$$y_{ex}\left(x\right)=e^{-x}+{\displaystyle \frac{2e^x}{3-e^2}}.$$

The results for this example are shown in

Table 5. Results of solving the integral equation in Example 5.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$0.544$	$0.544$	$0.544$	$0.544$	$1.27E-07$	$3.01E-11$	$1.90E-18$
0.1	$0.401$	$0.401$	$0.401$	$0.401$	$1.40E-07$	$3.33E-11$	$2.10E-18$
0.2	$0.262$	$0.262$	$0.262$	$0.262$	$1.55E-07$	$3.68E-11$	$2.32E-18$
0.3	$0.126$	$0.126$	$0.126$	$0.126$	$1.71E-07$	$4.07E-11$	$2.57E-18$
0.4	$-0.009$	$-0.009$	$-0.009$	$-0.009$	$1.89E-07$	$4.50E-11$	$2.84E-18$
0.5	$-0.145$	$-0.145$	$-0.145$	$-0.145$	$2.09E-07$	$4.97E-11$	$3.14E-18$
0.6	$-0.281$	$-0.281$	$-0.281$	$-0.281$	$2.31E-07$	$5.49E-11$	$3.47E-18$
0.7	$-0.421$	$-0.421$	$-0.421$	$-0.421$	$2.55E-07$	$6.07E-11$	$3.83E-18$
0.8	$-0.565$	$-0.565$	$-0.565$	$-0.565$	$2.82E-07$	$6.71E-11$	$4.24E-18$
0.9	$-0.714$	$-0.714$	$-0.714$	$-0.714$	$3.11E-07$	$7.42E-11$	$4.68E-18$
1.0	$-0.871$	$-0.871$	$-0.871$	$-0.871$	$3.44E-07$	$8.19E-11$	$5.17E-18$
				MAE	$3.44E-07$	$8.19E-11$	$5.17E-18$

.

The authors of [33] solved integral Equation (58) using the integral method of mean values and obtained the result with the maximum absolute error $E_m=2.0\times {10}^{-5}$ at $N=11$.

Our method gave the result with the value of maximum absolute error $E_m=3.44\times {10}^{-7}-5.17\times {10}^{-18}$ at $m=\overline{2,8}$.

Example 6.

In (50) $K(x,s)=e^{x-s-12},f\left(x\right)=\cos \left(x\right)-{\displaystyle \frac{e^{-x-12}}{2}}[e\times \sin \left(1\right)+e\times \cos \left(1\right)-1],\lambda =-1,a=0,b=1$. Then integral Equation (50) will take the following form:

$$y\left(x\right)+{\int }_0^1{e}^{x-s-12}\times y\left(s\right)ds=\cos \left(x\right)-{\displaystyle \frac{e^{-x-12}}{2}}\times \left(e\times \sin \left(1\right)+e\times \cos \left(1\right)-1\right),x\in [0,1].$$

(59)

The exact solution of integral Equation (59) is as follows:

$$y_{ex}\left(x\right)=\cos \left(x\right).$$

The results for this example are given in

Table 6. Results of solving the integral equation in Example 6.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$1.000$	$1.000$	$1.000$	$1.000$	$4.02E-13$	$9.56E-17$	$6.04E-24$
0.1	$0.995$	$0.995$	$0.995$	$0.995$	$3.63E-13$	$8.65E-17$	$5.46E-24$
0.2	$0.980$	$0.980$	$0.980$	$0.980$	$3.29E-13$	$7.83E-17$	$4.94E-24$
0.3	$0.955$	$0.955$	$0.955$	$0.955$	$2.98E-13$	$7.08E-17$	$4.47E-24$
0.4	$0.921$	$0.921$	$0.921$	$0.921$	$2.69E-13$	$6.41E-17$	$4.05E-24$
0.5	$0.878$	$0.878$	$0.878$	$0.878$	$2.44E-13$	$5.80E-17$	$3.66E-24$
0.6	$0.825$	$0.825$	$0.825$	$0.825$	$2.20E-13$	$5.25E-17$	$3.31E-24$
0.7	$0.765$	$0.765$	$0.765$	$0.765$	$1.99E-13$	$4.75E-17$	$3.00E-24$
0.8	$0.697$	$0.697$	$0.697$	$0.697$	$1.80E-13$	$4.30E-17$	$2.71E-24$
0.9	$0.622$	$0.622$	$0.622$	$0.622$	$1.63E-13$	$3.89E-17$	$2.45E-24$
1.0	$0.540$	$0.540$	$0.540$	$0.540$	$1.48E-13$	$3.52E-17$	$2.22E-24$
				MAE	$4.02E-13$	$9.56E-17$	$6.04E-24$

.

The authors of [34] solved integral Equation (59) by applying a new type of spline function of fractional order and obtained the result with the maximum absolute error $E_m=3.1\times {10}^{-6}$ at $N=11$ and $n=2$. Our method gave the result with the value of maximum absolute error $E_m=7.06\times {10}^{-9}-6.04\times {10}^{-24}$ at $m=\overline{1,8}$.

Example 7.

In (50) $K(x,s)=e^{2x-{\displaystyle \frac{5s}{3}}},f\left(x\right)=e^{2x+{\displaystyle \frac{1}{3}}},\lambda =-{\displaystyle \frac{1}{3}},a=0,b=1$. Then integral Equation (50) will take the following form:

$$y\left(x\right)+{\displaystyle \frac{1}{3}}{\int }_0^1{e}^{2x-{\displaystyle \frac{5s}{3}}}\times y\left(s\right)ds=e^{2x+{\displaystyle \frac{1}{3}}},x\in [0,1].$$

(60)

The exact solution of integral Equation (60) is as follows:

$$y_{ex}\left(x\right)=e^{2x}.$$

The results for this example are given in

Table 7. Results of solving the integral equation in Example 7.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$1.000$	$1.000$	$1.000$	$1.000$	$1.16E-06$	$1.10E-09$	$1.11E-15$
0.1	$1.221$	$1.221$	$1.221$	$1.221$	$1.41E-06$	$1.34E-09$	$1.36E-15$
0.2	$1.492$	$1.492$	$1.492$	$1.492$	$1.72E-06$	$1.64E-09$	$1.66E-15$
0.3	$1.822$	$1.822$	$1.822$	$1.822$	$2.11E-06$	$2.01E-09$	$2.03E-15$
0.4	$2.226$	$2.226$	$2.226$	$2.226$	$2.57E-06$	$2.45E-09$	$2.48E-15$
0.5	$2.718$	$2.718$	$2.718$	$2.718$	$3.14E-06$	$2.99E-09$	$3.02E-15$
0.6	$3.320$	$3.320$	$3.320$	$3.320$	$3.84E-06$	$3.66E-09$	$3.69E-15$
0.7	$4.055$	$4.055$	$4.055$	$4.055$	$4.69E-06$	$4.47E-09$	$4.51E-15$
0.8	$4.953$	$4.953$	$4.953$	$4.953$	$5.73E-06$	$5.45E-09$	$5.51E-15$
0.9	$6.050$	$6.050$	$6.050$	$6.050$	$6.99E-06$	$6.66E-09$	$6.73E-15$
1.0	$7.389$	$7.389$	$7.389$	$7.389$	$8.54E-06$	$8.14E-09$	$8.22E-15$
				MAE	$8.54E-06$	$8.14E-09$	$8.22E-15$

.

The authors of [35] solved integral Equation (60) based on a special representation of vector forms of triangular functions and obtained the result with the maximum absolute error $E_m=6.4\times {10}^{-7}$ at $m=1024$.

The authors of [39] using a combination of Taylor series and block-plus functions solved the same integral Equation (60). They obtained the result with the maximum absolute error $E_m=4.6\times {10}^{-4}$ at $N=80$.

The authors of [40] solved integral Equation (60) using Chebyshev polynomial approximation and obtained the result with the maximum absolute error $E_m=0.49\times {10}^{-4}$ at $N=10$.

With the scheme based on Legendre polynomials and Legendre wavelets, the authors of [43] solved integral Equation (60) and obtained the result with the maximum absolute error $E_m={10}^{-4}$ at $N=11,M=8$, and $k=3$.

Our method gave the result with the maximum absolute error from $E_m=8.14\times {10}^{-9}$ to $E_m=8.22\times {10}^{-15}$ at $m=\overline{4,8}$.

Example 8.

In (50) $K(x,s)=x\times s,f\left(x\right)=e^x-x,\lambda =1,a=0,b=1$. Then integral Equation (50) will take the following form:

$$y\left(x\right)-{\int }_0^{1}x\times s\times y\left(s\right)ds=e^x-x,x\in [0,1]$$

(61)

The exact solution of integral Equation (61) is as follows:

$$y_{ex}\left(x\right)=e^x.$$

The results for this example are shown in

Table 8. Results of solving the integral equation in Example 8.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$-0.186$	$-0.186$	$-0.186$	$-0.186$	$6.00E-32$	$2.30E-31$	$2.70E-31$
0.1	$-0.205$	$-0.205$	$-0.205$	$-0.205$	$0.00E+00$	$6.00E-32$	$1.00E-31$
0.2	$-0.227$	$-0.227$	$-0.227$	$-0.227$	$1.00E-32$	$2.90E-31$	$3.00E-31$
0.3	$-0.250$	$-0.250$	$-0.250$	$-0.250$	$2.20E-31$	$2.60E-31$	$1.80E-31$
0.4	$-0.277$	$-0.277$	$-0.277$	$-0.277$	$4.90E-31$	$6.50E-31$	$6.60E-31$
0.5	$-0.306$	$-0.306$	$-0.306$	$-0.306$	$3.40E-31$	$5.30E-31$	$6.10E-31$
0.6	$-0.338$	$-0.338$	$-0.338$	$-0.338$	$3.30E-31$	$5.20E-31$	$6.60E-31$
0.7	$-0.374$	$-0.374$	$-0.374$	$-0.374$	$1.20E-31$	$5.00E-31$	$4.90E-31$
0.8	$-0.413$	$-0.413$	$-0.413$	$-0.413$	$1.20E-31$	$6.00E-32$	$1.70E-31$
0.9	$-0.456$	$-0.456$	$-0.456$	$-0.456$	$5.00E-32$	$5.00E-32$	$1.30E-31$
1.0	$-0.504$	$-0.504$	$-0.504$	$-0.504$	$6.00E-32$	$8.00E-32$	$6.00E-32$
				MAE	$4.90E-31$	$6.50E-31$	$6.60E-31$

.

The author [37] solved integral Equation (61) using the Pell–Lucas series method and obtained the result with the maximum absolute error $E_m=7.5\times {10}^{-8}$ at $N=11$.

Our method gave the result with the maximum absolute error $E_m=3.57\times {10}^{-11}-2.25\times {10}^{-18}$ at $m=\overline{4,8}$.

Example 9.

In (50) $K(x,s)=e^{{\displaystyle \frac{x-s}{2}}},f\left(x\right)=x,\lambda ={\displaystyle \frac{1}{2}},a=0,b=1$. Then the integral Equation (50) will take the following form:

$$y\left(x\right)-{\displaystyle \frac{1}{2}}{\int }_0^1{e}^{{\displaystyle \frac{x-s}{2}}}y\left(s\right)ds=x,x\in [0,1].$$

(62)

The exact solution of integral Equation (62) is as follows:

$$y_{ex}\left(x\right)=x+(4\times \sqrt{e}-6)\times e^{{\displaystyle \frac{x-1}{2}}}.$$

The results for this example are shown in

Table 9. Results of solving the integral equation in Example 9.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$1.000$	$1.000$	$1.000$	$1.000$	$0.00E+00$	$0.00E+00$	$0.00E+00$
0.1	$1.105$	$1.105$	$1.105$	$1.105$	$1.50E-08$	$3.57E-12$	$2.25E-19$
0.2	$1.221$	$1.221$	$1.221$	$1.221$	$3.00E-08$	$7.14E-12$	$4.51E-19$
0.3	$1.350$	$1.350$	$1.350$	$1.350$	$4.50E-08$	$1.07E-11$	$6.76E-19$
0.4	$1.492$	$1.492$	$1.492$	$1.492$	$6.00E-08$	$1.43E-11$	$9.01E-19$
0.5	$1.649$	$1.649$	$1.649$	$1.649$	$7.50E-08$	$1.78E-11$	$1.13E-18$
0.6	$1.822$	$1.822$	$1.822$	$1.822$	$8.99E-08$	$2.14E-11$	$1.35E-18$
0.7	$2.014$	$2.014$	$2.014$	$2.014$	$1.05E-07$	$2.50E-11$	$1.58E-18$
0.8	$2.226$	$2.226$	$2.226$	$2.226$	$1.20E-07$	$2.86E-11$	$1.80E-18$
0.9	$2.460$	$2.460$	$2.460$	$2.460$	$1.35E-07$	$3.21E-11$	$2.03E-18$
1.0	$2.718$	$2.718$	$2.718$	$2.718$	$1.50E-07$	$3.57E-11$	$2.25E-18$
				MAE	$1.50E-07$	$3.57E-11$	$2.25E-18$

.

The authors of [38] solved integral Equation (62) using the Taylor series expansion method and obtained the result with the maximum absolute error $E_m=8.88\times {10}^{-16}$ in $N=11$ and $p=12$.

Our method gave the result with the maximum absolute error $E_m=3.84\times {10}^{-17}-2.43\times {10}^{-21}$ at $m=\overline{6,8}$.

Example 10.

In (50) $K(x,s)=x+s,f\left(x\right)=e^x+(1-e)\times x-1,\lambda =1,a=0,b=1$. Then integral Equation (50) will take the following form:

$$y\left(x\right)-{\int }_0^1(x+s)\times y\left(s\right)ds=e^x+(1-e)\times x-1,x\in [0,1].$$

(63)

The exact solution of integral Equation (63) is as follows:

$$y_{ex}\left(x\right)=e^x.$$

The results for this example are shown in

Table 10. Results of solving the integral equation in Example 10.

${\mathit{x}}_{\mathit{i}}$	ES	OQF			Absolute Error
		$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$	$\mathit{m}=\mathbf{2}$	$\mathit{m}=\mathbf{4}$	$\mathit{m}=\mathbf{8}$
0.0	$0.361$	$0.361$	$0.361$	$0.361$	$6.26E-09$	$3.73E-13$	$1.47E-21$
0.1	$0.479$	$0.479$	$0.479$	$0.479$	$6.59E-09$	$3.92E-13$	$1.55E-21$
0.2	$0.599$	$0.599$	$0.599$	$0.599$	$6.92E-09$	$4.12E-13$	$1.63E-21$
0.3	$0.719$	$0.719$	$0.719$	$0.719$	$7.28E-09$	$4.33E-13$	$1.71E-21$
0.4	$0.841$	$0.841$	$0.841$	$0.841$	$7.65E-09$	$4.55E-13$	$1.80E-21$
0.5	$0.963$	$0.963$	$0.963$	$0.963$	$8.04E-09$	$4.79E-13$	$1.89E-21$
0.6	$1.087$	$1.087$	$1.087$	$1.087$	$8.46E-09$	$5.03E-13$	$1.99E-21$
0.7	$1.212$	$1.212$	$1.212$	$1.212$	$8.89E-09$	$5.29E-13$	$2.09E-21$
0.8	$1.338$	$1.338$	$1.338$	$1.338$	$9.35E-09$	$5.56E-13$	$2.19E-21$
0.9	$1.466$	$1.466$	$1.466$	$1.466$	$9.83E-09$	$5.85E-13$	$2.31E-21$
1.0	$1.595$	$1.595$	$1.595$	$1.595$	$1.03E-08$	$6.15E-13$	$2.43E-21$
				MAE	$1.03E-08$	$6.15E-13$	$2.43E-21$

.

The authors of [38] solved integral Equation (63) and obtained the result with the maximum absolute error $E_m=5.6\times {10}^{-9}$ in $N=11$ and $p=8$.

The authors of [39], in addition to integral Equation (60), solved integral Equation (63) and obtained the result with the maximum absolute error $E_m=2.84\times {10}^{-6}$ at $N=80$.

The authors of [46] solved integral Equation (63) using general Legendre wavelets and obtained the result with the highest absolute error $E_m=0.7\times {10}^{-13}$ in $M=11$ and $N=4$. Our method gave the result with the maximum absolute error $E_m=8.82\times {10}^{-18}$ at $m=8$.

8. Conclusions

This work focuses on solving the optimization problem for weighted quadrature formulas with derivatives in Sobolev spaces. Through examples involving linear Fredholm integral equations of the second kind, the theoretical soundness of the constructed quadrature formulas is demonstrated by comparing results with those obtained by other researchers.

The main results of this study are as follows:

• The squared norm of the error functional for the quadrature formula under consideration was computed using the extremal function;
• This norm was minimized with respect to the quadrature formula coefficients, leading to a system of linear algebraic equations for determining the optimal coefficients;
• The uniqueness of the solutions to this system was established;
• An algorithm was developed to solve this system, enabling the determination of optimal coefficients for weighted quadrature formulas with derivatives;
• A procedure was provided for applying these weighted quadrature formulas with derivatives to approximate solutions of linear Fredholm integral equations of the second kind;
• Simplified expressions for the optimal coefficients of quadrature formulas with derivatives were derived for the spaces $L_2^{\left(1\right)}(0,1)$, $L_2^{\left(2\right)}(0,1)$ and $L_2^{\left(3\right)}(0,1)$;
• Finally, numerical results were compared against those reported by other authors to validate the approach.