The numerical solution of singular integro-differential equations in Hölder spaces

Caraus Iurie

Department of Mathematics and Informatics, Moldova State University
Mateevici 60, str., Chisinau, Moldova, MD-2009

caraush@usm.md

Contributed talk

Let $\Gamma$ be a smooth Jordan border limiting the one-spanned area $%F^{+},$ containing a point $t=0,$ $F^{-}=C\setminus \{F^{+}\cup \Gamma \}.$ Let $z=\psi (w)$ be a function, mapping conformably and unambiguously the border $\{\vert w\vert>1\}$ on the surface $F^{-}$, so that $\psi (\infty )=\infty ,\psi ^{(\prime )}(\infty )>0.$We denote $H_{\beta }(\Gamma )$ the spaces of functions satisfying on $\Gamma$ the Hölder condition with some parameter $\beta$ $(0<\beta <1).$ In the complex space $H_{\beta }(\Gamma )$ of functions $g(t)$ with norm

$$\left\Vert g\right\Vert _{\beta }=\left\Vert g\right\Vert _... ...\beta }},\quad t^{^{\prime }},t^{^{\prime \prime }}\in \Gamma$$

we will consider the singular integro-differential equations (SIDE)

$$(Mx\equiv )\sum\limits_{r=0}^{q}\Big[ \tilde{A_r}(t)x^{(r)}... ...au )\cdot x^{(r)}(\tau )d\tau \Big]=f(t),\quad t\in \Gamma ,$$ (1)

where $\tilde{A}_{r}(t),\tilde{B}_{r}(t)$ and $K_{r}(t,\tau )$ $(r=\overline{%0,q})$ and $f(t)$are given functions which belong to $H_{\beta }(\Gamma )$; $%x^{(0)}(t)=x(t)$ is the required functions; $x^{(r)}(t)=\frac{d^{r}x(t)}{%dt^{r}}$ $(r=\overline{1,q});q$ is a natural number which belong to $%H_{\beta }(\Gamma )$. We search the solution of equation (1) in the class of functions, satisfying the condition

$$\frac{1}{2\pi i}\int\limits_{\Gamma }x(\tau )\tau ^{-k-1}d\tau =0,\quad k=%\overline{0,q-1}.$$ (2)

We introduce the denomination ''the problem (1)-(2)'' for the SIDE (1) together with the conditions (2). We search the approximate solution of the problem (1)-(2) in the form

$$x_{n}(t)=\sum_{k=0}^{n}\xi _{k}^{(n)}t^{k+q}+\sum\limits_{k=-n}^{-1}\xi _{k}^{(n)}t^{k},\quad t\in \Gamma ,$$ (3)

where $\xi _{k}^{(n)}=\xi _{k}$ $(k=\overline{-n,n})$ are unknowns numbers; we will note that the function $x_{n}(t),$ constructed by formula (3), obviously, satisfies the condition (2). We have elaborated the numerical schemes of the collocations method and quadrature method for approximate solution of the SIDE. We investigate the case when the equations defined on the arbitrary smooth boundaries. The theoretical foundation of these methods have been obtained when the knots of discretization forme a system of Feyer knots, and their convergence is given in Hölder spaces.