Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications

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, optimization of computational algorithms. 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. Then the error functional [1] and [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 in finding such a functional (1) whose norm in the space $B^{*}$ is minimal.

Studies of optimal and asymptotic cubature formulas are found in [1] - [11]. Optimization studies of quadrature formulas are presented in [12] - [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, a number of new results have been obtained on the construction and their error estimates of optimal quadrature formulas for approximate computation of regular, singular, and integrals from rapidly oscillating functions using the Sobolev method. These results can be found, for example, in [15] - [27].

In this paper, the construction of composite optimal quadrature formulas with weight in Sobolev space is studied by 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. Minimizing this norm by coefficients, the system of algebraic equations is obtained. The uniqueness of the solutions of the obtained system is proved. Using this algorithm, the optimal coefficients of quadrature formulas with a weight function are found.

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 approximate solutions of specific linear Fredholm integral equations of the second kind are found by the constructed optimal quadrature formula, and they are compared with the results of [29] - [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 $L_2^{\left(m\right)}(0,1)$ - the space of functions whose m - th generalized derivative sums to square on the interval [0,1], $p\left(x\right)$-the weight function whose

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

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

The error of the quadrature formula is the difference of

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

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)

${\mathcal{E}}_{[0,1]}$ - index of the segment [0,1], $\delta \left(x\right)$ - Dirac 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

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

(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 in finding the value of

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

(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

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

(7)

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

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

$P_{m-1}\left(x\right)$ - 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] and [2]. For any $0\le t\le m-1$ theorem 1 is proved in [2].

Since $L_2^{\left(m\right)}(0,1)$ - Hilbert space, the norm of the error functional ${\ell }_N^{\left(t\right)}$ and the function ${\psi }_{\ell }\left(x\right)$ are related by the relation

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

(8)

In addition, there is an equality

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

(9)

Substituting the extremal function defined by formula (7) into equality (9) and considering (5), after some calculations for the square of the norm of the error functional (4) of the quadrature formula (3) we obtain

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

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

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

(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)

equivalent to the accuracy conditions (5) of the quadrature formula on polynomials of degree lower 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 under which the quadratic function $F\left(C\right)$ on the set of vectors

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

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

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

(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 system (13) - (14), the unknowns are $C_{\beta }^{\left(\nu \right)}$ and ${\lambda }_k$. The solution of this system is the stationary point of the function $F(C,\lambda )$, which we denote as ${\stackrel{\circ }{C}}_{\beta }^{\left(\nu \right)}$ and ${\stackrel{\circ }{\lambda }}_k$. From the theory of conditional extremal, we know a sufficient condition under which this solution gives a conditional minimum of $F\left(C\right)$ on the manifold (5). It consists in positive definiteness of the quadratic form

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

(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 the 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 lying in subspace $SC=0$, the function $\Phi \left(C\right)$ is strictly positive.

Proof 

(Proof of lemma 1.). From the definition of the Lagrange function $F(C,\lambda )$ and from equality (15) it 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 Eq.

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

(20)

The solution of Equation (20) has the form

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

(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 $\varphi \left(C\right)$

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

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

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

Proof of Lemma 2.

Let us write the homogeneous system corresponding to the system (13) - (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’s denote by G the matrix of quadratic form (18), and write the homogeneous system (22) - (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 the 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 ${\mu }_{\overline{C}}\left(x\right)$:

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

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

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

The system of equations (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

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

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

Theorem 2.

Let the error functional of the quadrature formula (3) ${\ell }_N^{\left(t\right)}\left(x\right)$ be defined in the space $L_2^{\left(m\right)}(0,1)$, i.e., its values for all polynomials of degree $m-1$ are zero and optimal, i.e., among all functionals of the form (4) with given nodes $x_{\beta }$ it has the smallest norm in $L_2^{\left(m\right)}(0,1)$. Then there exists a solution ${\psi }_{\ell }\left(x\right)$ of Eq.

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

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 [28], i.e. it is proved that the extremal function for the error functional goes to zero at nodes $x_{\beta }$.

3. On One Method of Construction of Weighted Optimal quadrature formulas with derivatives

In this section we will 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 Eq.

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

and can be written in the form

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

(29)

where

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

$P_{m-1}\left(x\right)$ - 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

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

(30)

The function ${\psi }_{\ell }\left(x\right)$ is expressed, as we have seen, by formula (29), so calculating the square of the norm of the error functional using (30), we obtain

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

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

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

(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 of Construction of Weighted Optimal Quadrature formulas derivative formulas

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

Our method for constructing optimal quadrature formulas with derivatives is as follows. First, for $m=1$, i.e., in the space $L_2^{\left(1\right)}(0,1)$ minimizing the square of the norm of the error functional (31) by the coefficients $C_{\beta }^{\left(0\right)}(\beta =\overline{0,N})$, under conditions (28), we obtain the following system for finding $C_{\beta }^{\left(0\right)}$:

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

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

(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 $m=1$. The system (32) - (33) is solved in [2], 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 linear integral equations is given in [40].

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 derivative formulas

We now solve the system of linear algebraic equations (34) - (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 the system (34) - (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 ${\lambda }_k$ - constant, $F_{k\beta }$ 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 $(h\beta -h\gamma )$. This kind of equations in the continuous case, where instead of the sum there are integrals, are called Wiener - Hopf equations. As is usually done 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 the system (37) - (38) is written in the form of convolution equations

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

(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 the system (39) - (41). For this purpose, instead of ${\stackrel{\circ }{C}}^{\left(k\right)}\left[\beta \right]$ we introduce the following function

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

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

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

$a_k^{-}$ - unknown.

Let $\beta \ge N$, then

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

$a_k^{+}$ - 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, so 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]:

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

(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 $\beta =\overline{0,N}$:

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

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

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

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

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 $B_{k+1}$ - the Bernoulli numbers.

Corollary 1 shows that the optimal coefficients are the coefficients of the well-known Euler - Maclorean quadrature formula. The optimality of the Euler - Maclorean quadrature formula in the space $L_2^{\left(m\right)}(0,1)$ is proved in [2,9].

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

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

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

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

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

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

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

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

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

(50)

where $K(x,s)$ - the kernel of the integral equation, $f\left(x\right)$ - the right-hand side, $\lambda$ - the parameter of the integral equation, $y\left(x\right)$ - an unknown function to be determined.

Several numerical methods have been described for the numerical approximation of the solution of (50) (collocation methods, projection methods, Galerkin methods, etc.) and have been extensively investigated in terms of stability and convergence in suitable function spaces, also according to the smoothness properties of the kernel K and the right-hand side f; (see. [48]-[66]).

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

In the system of equations (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, we have

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

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

(52)

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

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

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

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

(53)

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

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 are to solve integral equations by the quadrature method using the optimal quadrature formula with derivatives at $m=2,4,8$. Compare the results with the exact solution and the results of foreign researchers using the absolute error:

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

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

It should be noted, according to the above algorithm, a program in Maple language has been made. All calculations are performed using 32 significant digits.

Example 1. In (50) $K(x,s)=-x·(e^{x·s}-1),f\left(x\right)=e^x-x,\lambda =1,a=0,b=1$. Then the integral equation (6.1) will take the following form:

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

(54)

Exact solution of the integral Equation (54):

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

The results for this example are summarized in Table 1.

               Table 1

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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-Exact solution, OQF-Optimal quadrature formulas, MAE-Maximum absolute error.

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

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 $E_m=4.44·{10}^{-16}$ at the number of integration intervals $N=20$.

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

Example 2. In (50), $K(x,s)=x·s,f\left(x\right)=cos\left(x\right),\lambda =1,a=0,b=1$. Then the integral Equation (50) takes the following form:

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

(55)

Exact solution of the integral Equation (55):

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

The results for this example are shown in Table 2.

               Table 2

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 $E_m=2.0·{10}^{-5}$ at $N=11,n=10$. Our method gave the result with the value of maximum absolute error $E_m=1.62·{10}^{-8}-2.43·{10}^{-19}$ 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 the 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)

Exact solution of the integral Equation (56):

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

The results for this example are given in Table 3.

                 Table 3

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 the integral Equation (56). They obtained the result with the maximum absolute error $E_m=8.88·{10}^{-16}$ at $n=3$.

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

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

Example 4. In (50) $K(x,s)=x·e^s,f\left(x\right)=e^{-x},\lambda =1,a=0,b=1$. Then the integral Equation (50) will take the following form:

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

(57)

Exact solution of the integral Equation (57):

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

The results for this example are given in Table 4.

                 Table 4

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 (57) integral equation, the author [32] of the paper applied polynomial method using Boo-Baker polynomials and obtained the result with maximum absolute error $E_m=7.1·{10}^{-3}$ 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 $E_m=2.0·{10}^{-3}$ at $N=11$ for $n=2$.

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

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

Our method gave the result with the value of maximum absolute error $E_m=2.09·{10}^{-18}$ 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 (6.1) will take the following form:

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

(58)

Exact solution of the integral Equation (58):

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

The results for this example are shown in Table 5.

                 Table 5

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 the integral Equation (58) using the integral method of mean values and obtained the result with the maximum absolute error $E_m=2.0·{10}^{-5}$ at $N=11$.

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

Example 6. In (50) $K(x,s)=e^{x-s-12},f\left(x\right)=cos\left(x\right)-{\displaystyle \frac{e^{-x-12}}{2}}[e·sin\left(1\right)+e·cos\left(1\right)-1],\lambda =1,a=0,b=1$. Then the integral Equation (50) will take the following form:

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

(59)

Exact solution of the integral Equation (59):

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

The results for this example are given in Table 6.

                  Table 6

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 the integral Equation (59) by applying a new type of spline function of fractional order and obtained the result with the maximum absolute error $E_m=3.1·{10}^{-6}$ at $N=11$ and $n=2$. Our method gave the result with the value of maximum absolute error $E_m=7.06·{10}^{-9}-6.04·{10}^{-24}$ at $m=\overline{1,8}$.

Example 7. In (50) $K(x,s)=e^{2x-{\displaystyle \frac{5s}{3}}},f\left(x\right)=e^{2x+{\displaystyle \frac{1}{3}}},\lambda =1,a=0,b=1$. Then the integral Equation (50) will take the following form:

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

(60)

Exact solution of the integral Equation (60):

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

The results for this example are given in Table 7.

                  Table 7

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 the integral Equation (60) based on a special representation of vector forms of triangular functions and obtained the result with the maximum absolute error $E_m=6.4·{10}^{-7}$ 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 $E_m=4.6·{10}^{-4}$ at $N=80$.

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

With the scheme based on Legendre polynomials and Legendre wavelets, the authors of [43] solved the 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 $E_m=8.14·{10}^{-9}$ to $E_m=8.22·{10}^{-15}$ at $m=\overline{4,8}$.

Example 8. In (50) $K(x,s)=x·s,f\left(x\right)=e^x-x,\lambda =1,a=0,b=1$. Then the integral Equation (50) will take the following form:

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

(61)

Exact solution of the integral Equation (61):

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

The results for this example are shown in Table 8.

                 Table 8

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 the integral Equation (61) using the Pell-Lucas series method and obtained the result with the maximum absolute error $E_m=7.5·{10}^{-8}$ at $N=11$.

Our method gave the result with the maximum absolute error $E_m=3.57·{10}^{-11}-2.25·{10}^{-18}$ 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)

Exact solution of the integral Equation (62):

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

The results for this example are shown in Table 9.

                 Table 9

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 the integral Equation (62) using the Taylor series expansion method and obtained the result with the maximum absolute error $E_m=8.88·{10}^{-16}$ in $N=11$ and $p=12$.

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

Example 10. In (50) $K(x,s)=x+s,f\left(x\right)=e^x+(1-e)·x-1,\lambda =1,a=0,b=1$. Then the integral Equation (50) will take the following form:

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

(63)

Exact solution of the integral Equation (63):

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

The results for this example are shown in Table 10.

                 Table 10

$x_i$	ES	OQF			Absolute error
		$m=2$	$m=4$	$m=8$	$m=2$	$m=4$	$m=8$
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 the integral Equation (63) and obtained the result with the maximum absolute error $E_m=5.6·{10}^{-9}$ in $N=11$ and $p=8$.

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

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

8. Conclusion

The present work is devoted to the solution of the optimization problem of weight quadrature formulas with derivatives in the Sobolev space and to determine by means of examples on linear Fredholm integral equations of the second kind the theoretical validity of the constructed quadrature formula comparing the results of works performed by other authors.

The main results of the study are as follows:

- the square of the norm of the error functional of the considered quadrature formula was calculated using the extremal function;

- minimized this norm by the coefficients of the quadrature formula and obtained a system of linear algebraic equations for finding the optimal coefficients of the quadrature formula;

- the singularity of solutions of this system is proved;

- an algorithm for the solution of this system is constructed with the help of which the optimal coefficients of weight quadrature formulas with derivatives are found;

- an algorithm of application of weight quadrature formulas with derivatives for an approximate solution of linear Fredholm integral equations of the second kind is given;

- simplified formulas for the optimal coefficients of the quadrature formula with derivatives in the spaces $L_2^{\left(1\right)}(0,1),L_2^{\left(2\right)}(0,1)$ and $L_2^{\left(3\right)}(0,1)$ were obtained;

- in the end, the numerical results are compared with the results of other authors.