Richardson extrapolation for the discrete iterated modified projection solution (Q2192578)

The paper is devoted to the study of the following nonlinear Urysohn integral equation \[ x-\mathcal{K}(x)=f, \] where \(\mathcal{K}\) is a compact operator from \(L^{\infty}([0,1])\) to \(C([0,1])\) which is defined as follows \[ \mathcal{K}(x)(s)=\int\limits_{0}^{1}k(s,t,x(t)) \, dt, \quad s\in[0,1]. \] Some projection methods associated with a sequence of interpolatory projections are used. An asymptotic expansion for the discrete iterated modified projection solution is also obtained and the Richardson extrapolation to improve the order of convergence is used. In conclusion, the authors suggest some numerical results confirming that the computed orders of convergence match well with the theoretical orders of convergence and that the extrapolated solution improves upon the discrete iterated modified projection solution.