TY - JOUR
T1 - Asymptotic Error Analysis for the Discrete Iterated Galerkin Solution of Urysohn Integral Equations with Green’s Kernels
AU - Rakshit , Gobinda
JO - Advances in Applied Mathematics and Mechanics
VL - 1
SP - 175
EP - 206
PY - 2024
DA - 2024/12
SN - 17
DO - http://doi.org/10.4208/aamm.OA-2023-0004
UR - https://global-sci.org/intro/article_detail/aamm/23598.html
KW - Urysohn integral equation, Green’s kernel, iterated Galerkin method, Nyström approximation, Richardson extrapolation.
AB - <p style="text-align: justify;">Consider a Urysohn integral equation $x−\mathcal{K}(x)= f,$ where $f$ and the integral
operator $\mathcal{K}$ with kernel of the type of Green’s function are given. In the computation
of approximate solutions of the given integral equation by Galerkin method, all the
integrals are needed to be evaluated by some numerical integration formula. This
gives rise to the discrete version of the Galerkin method. For $r≥1,$ a space of piecewise
polynomials of degree $≤ r−1$ with respect to a uniform partition is chosen to be the
approximating space. For the appropriate choice of a numerical integration formula,
an asymptotic series expansion of the discrete iterated Galerkin solution is obtained
at the above partition points. Richardson extrapolation is used to improve the order
of convergence. Using this method we can restore the rate of convergence when the
error is measured in the continuous case. Numerical examples are given to illustrate
this theory.</p>