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On the application of linear methods to polynomial approximation of
solutions of ordinary differential equations and Hammerstein
integral equations
V. K. Dzyadyk
Abstract:
Starting from known linear polynomial operators
$U_n(\psi;x)$ which generate good approximations to continuous functions
$\psi(x)$, the author proposes a method which for a given right-hand side of the equation
\begin{equation}
y'=f(x,y)
\tag{1}
\end{equation}
and given initial conditions enables us to construct polynomials
$y_n(x)=y_n(U_n;f;x)$ approximating to the unknown solution of the equation (1) with essentially the same precision as these operators
$U_n$ would yield if the solution were given. More precisely, it is shown in this paper that
$|y(x)-y_n(U_n;f;x)|\leqslant(1+\alpha_n)\cdot C\|y(x)-U_n(y;x)\|$,
$C=\operatorname{const}$,
$\alpha_n\downarrow0$,
and effective upper bounds are placed on the quantities
$C$ and
$\alpha_n$. The same procedure is used also for the polynomial approximation of the solutions of
$k$-th order equations with
$k\geqslant2$, systems of equations, Hamrnerstein integral equations and other integral equations.
UDC:
517.9
MSC: 41A10,
41A30,
45B05,
47A58,
34A45 Received: 29.09.1969
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