Abstract
A popular class of methods for solving weakly singular integral equations is the class of piecewise polynomial collocation methods. In order to implement those methods one has to compute exactly certain integrals that determine the linear system to be solved. Unfortunately those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids. In this paper fully discrete analogs of the collocation methods, where integrals are replaced by quadrature formulas, are considered, corresponding error estimates are derived.
Highlights
IntroductionK is the kernel of the integral operator and φ is a basis function corresponding to the collocation method
In many applications there arise integral equations of the form b y(t) = K(t, s)y(s) ds + f (t), (1.1)where the integration kernel K is not smooth across the diagonal t = s
A popular class of methods for solving such equations is the class of piecewise polynomial collocation methods using
Summary
K is the kernel of the integral operator and φ is a basis function corresponding to the collocation method Those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids (see [4]). It is of great practical and theoretical interest to consider methods (so called fully discrete methods), where the integrals are computed by quadrature formulas. Tamme of the error caused by the quadrature approximation of the system integrals and extend the results to other types of integral equations
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