A Collocation Method Based on the Central Part Interpolation for Integral Equations

Abstract

We study the integral equation

$$\displaystyle{ u(x) =\int \limits _{ 0}^{1}[a(x,y)\vert x - y\vert ^{-\nu } + b(x,y)]u(y)dy + f(x),\quad 0 \leq x \leq 1,\quad 0 <\nu < 1, }$$

(37.1)

where $f \in C[0,1] \cap C^{m}(0,1)$, $a,b \in C^{m}([0,1] \times (0,1))$, $m \in \mathbb{N} = \left \{1,2,...\right \}$. By C ^m(Ω) is meant the set of all m times continuously differentiable functions on Ω. By C[0, 1] is meant the Banach space of continuous functions $u: [0,1] \rightarrow \mathbb{R} = (-\infty,\infty )$ with the usual norm $\left \|u\right \|_{\infty } = \left \{\max \left \vert u(x)\right \vert: 0 \leq x \leq 1\right \}$.Denote by T the integral operator of equation (37.1):

$$\displaystyle{ (Tu)(x) =\int \limits _{ 0}^{1}[a(x,y)\vert x - y\vert ^{-\nu } + b(x,y)]u(y)dy\quad 0 \leq x \leq 1,\quad 0 <\nu < 1. }$$

(37.2)