Ufa Mathematical Journal
Volume 12, Number 1, pp. 43-55
On asymptotic convergence of polynomial collocation method for one class of
singular integro-differential equations
Fedotov A.I.
DOI:10.13108/2020-12-1-43
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Article on MathNetAbstact
Among the approximate methods for solving the operator equations,
the most used methods are collocation and Galerkin methods. Each of them has their own advantages
and disadvantages. For instance, Galerkin methods are used for the equations in Hilbert spaces. The estimates for the errors
of the solutions obtained by these methods have the order of the best approximations
of the exact solutions. However, Galerkin methods are not always constructive, as for their
implementation one needs to calculate integrals and this is not always possible to do explicitly.
Collocation methods are used for the equations in the spaces of continuous functions and thus are always
constructive. However, the estimates for the errors obtained by collocation methods are usually worse than
those of the best approximation of the exact solutions.
In the present paper, we justify a polynomial collocation method
for one class of singular integro-differential equations on an interval. For the justification, the technic of reducing
the polynomial collocation method to Galerkin method is used for the first time for such equations. This technique was first used by the author
to justify the polynomial collocation method for a wide class of periodic singular
integro-differential and pseudo-differential equations.
For the equations on a open interval, this approach is used for the first time. Also for the first time we prove that the interpolative Lagrange operator is bounded in the Sobolev spaces
$H_q^s$, $s>\tfrac{1}{2}$, with the Chebyshev weight function of the second kind. Exactly this result
gives an opportunity to show that in non-periodic the polynomial collocation
method provides the same convergence rate as the Galerkin method.