Example 5. Solve the equation

$$
x^{2}=\frac{2}{\pi} \int_{0}^{\pi / 2} \varphi(x \sin \theta) d \theta
$$
Solution. Equation (34) is a Schlömilch equation, where $f(x)=x^{2}$, and thus, $f(0)=0$. We find the derivative: $f^{\prime}(x)=2 x$. Applying formula (33), we find

$$
\varphi(x)=x \int_{0}^{\pi / 2} 2(x \sin \psi) d \psi=-\left.2 x^{2} \cos \psi\right|_{\psi=0} ^{\phi=\pi / 2}=2 x^{2}
$$

Answer: $\varphi(x)=2 x^{2}$.

By direct substitution of the function $\varphi(x)=2 x^{2}$ into equation (34), we verify that it is a solution.

## Problems for Independent Solution

Solve the following integral equations:

309. $1+x=\frac{2}{\pi} \int_{0}^{\pi / 2} \varphi(x \sin \theta) d \theta . \quad$ 310. $3 x^{2}+2 x^{3}=\frac{2}{\pi} \int_{0}^{\pi / 2} \varphi(x \cos \theta) d \theta$
310. $1+x^{2}=\frac{2}{\pi} \int_{0}^{\pi / 6} \varphi(x \sin 3 \theta) d \theta$.

## CHAPTER

## Approximate Methods for Solving Equations

## § 23. Replacing the Kernel of an Integral Equation with a Degenerate Kernel

Suppose we have an integral equation

$$
\varphi(x)=f(x)+\lambda \int_{a}^{b} K(x, t) \varphi(t) d t
$$

with an arbitrary kernel $K(x, t)$. The simplicity of finding the solution to an equation with a degenerate kernel (see §9) naturally leads to the idea of approximately replacing the given arbitrary kernel $K(x, t)$ with a degenerate kernel $L(x, t)$. In this case, the solution $\widetilde{\varphi}(x)$ of the new equation

$$
\tilde{\varphi}(x)=f_{1}(x)+\lambda \int_{a}^{b} L(x, t) \widetilde{\varphi}(t) d t
$$

is accepted as an approximation to the solution $\varphi(x)$ of the original equation (1). As a degenerate kernel $L(x, t)$, close to the given $K(x, t)$, one can take a segment of the Taylor series for the function $K(x, t)$, a segment of the Fourier series for $K(x, t)$ in any complete orthonormal system of functions $\left\{u_{n}(x)\right\}$ in $L_{2}(a, b)$, etc. We will indicate some error estimates in the solution (1) arising from the replacement of the given kernel with a degenerate one.

Let two kernels $L(x, t)$ and $K(x, t)$ be given and it is known that

$$
\int_{a}^{b}|K(x, t)-L(x, t)| d t0
$$

then the equation

$$
\varphi(x)=\lambda \int_{a}^{b} K(x, t) \varphi(t) d t+f(x)
$$

has a unique solution $\varphi(x)$ and the difference between this solution and the solution $\tilde{\varphi}(x)$ of the equation

$$
\tilde{\varphi}(x)=f_{1}(x)+\lambda \int_{a}^{b} L(x, t) \tilde{\varphi}(t) d t
$$

does not exceed

$$
|\varphi(x)-\widetilde{\varphi}(x)|<\frac{N|\lambda|(1+|\lambda| R)^{2} h}{1-|\lambda| h(1+|\lambda| R)}+\eta
$$

where $N$ is the upper bound of $|f(x)|$.

For a degenerate kernel $L(x, t)$, the resolvent $R_{L}(x, t ; \lambda)$ is found simply, namely, if

$$
L(x, t)=\sum_{k=1}^{n} X_{k}(x) T_{k}(t)
$$

then, setting

$$
\int_{a}^{b} X_{k}(x) T_{s}(x) d x=a_{s k}
$$

we obtain

$$
R_{L}(x, t ; \lambda)=\frac{D(x, t ; \lambda)}{D(\lambda)}
$$

where

$$
\begin{gathered}
D(x, t ; \lambda)=\left|\begin{array}{cccc}
0 & X_{1}(x) & \cdots & X_{n}(x) \\
T_{1}(t) & 1-\lambda a_{11} & \cdots & -\lambda a_{1 n} \\
\cdots \cdots \cdots & \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\
T_{n}(t) & -\lambda a_{n 1} & \cdots & 1-\lambda a_{n n}
\end{array}\right| \\
D(\lambda)=\left|\begin{array}{cccc}
1-\lambda a_{11} & -\lambda a_{12} & \cdots & -\lambda a_{1 n} \\
-\lambda a_{21} & 1-\lambda a_{22} & \cdots & -\lambda a_{2 n} \\
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots & \cdots \cdots \cdots \cdots & \cdots \cdots \\
-\lambda a_{n 1} & -\lambda a_{n 2} & \cdots & 1-\lambda a_{n n}
\end{array}\right|
\end{gathered}
$$

The roots of $D(\lambda)$ are the characteristic numbers of the kernel $L(x, t)$.

We will provide one more estimate $(\lambda=1)$. Let

$$
K(x, t)=L(x, t)+\Lambda(x, t)
$$

where $L(x, t)$ is a degenerate kernel, and $\Lambda(x, t)$ has a small norm in some metric. Further, let $R_{K}(x, t), R_{L}(x, t)$ be the resolvents of the kernels $K(x, t)$ and $L(x, t)$, respectively. $\|\Lambda\|,\left\|R_{K}\right\|,\left\|R_{L}\right\|-$ are the norms of the operators with the corresponding kernels. Then

$$
\|\varphi-\widetilde{\varphi}\| \leqslant\|\Lambda\| \cdot\left(1+\left\|R_{K}\right\|\right)\left(1+\left\|R_{L}\right\|\right) \cdot\|f\|,
$$

where the norm in formula (2) can be taken in any functional space. For the norm of the resolvent $R$ of any kernel $K(x, t)$, the estimate

$$
\|R\| \leqslant \frac{\|K\|}{1-|\lambda| \cdot\|K\|}
$$

holds. In the space $C(0,1)$ of continuous functions on the interval $[0,1]$

$$
\begin{aligned}
& \|K\|=\max _{0 \leqslant x \leqslant 1} \int_{0}^{1}|K(x, t)| d t \\
& \|f\|=\max _{0 \leqslant x \leqslant 1}|f(x)|
\end{aligned}
$$

In the space of functions summable with the square over $\Omega\{a \leqslant x$, $t \leqslant b\}$

$$
\begin{aligned}
& \|K\| \leqslant\left(\int_{a}^{b} \int_{a}^{b} K^{2}(x, t) d x d t\right)^{1 / 2} \\
& \|f\|=\left(\int_{a}^{b} f^{2}(x) d x\right)^{1 / 2}
\end{aligned}
$$