Approximate solution of Urysohn integral equations with non-smooth kernels

Abstract

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a kernel of the type of Green's function and defined on $L^\infty [0, 1]$. For $ r \geq 0$, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree $\leq r$ with respect to a quasi-uniform partition of $[0, 1]$ and consider an interpolatory projection at $r+1$ Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are $ r+1$ and $r + 2 + \min \{ r, 1 \}$, respectively. We show that the order of convergence in the iterated modified projection method is $4$ if $ r = 0$ and is $ 2 r + 3$ if $ r \geq 1$. This improvement in the order of convergence is achieved while retaining the size of the system of equations that needs to be solved, the same as in the case of the collocation method. Numerical results are given for specific examples.