Open Access
2016 Approximate solution of Urysohn integral equations with non-smooth kernels
Rekha P. Kulkarni, T.J. Nidhin
J. Integral Equations Applications 28(2): 221-261 (2016). DOI: 10.1216/JIE-2016-28-2-221


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.


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Rekha P. Kulkarni. T.J. Nidhin. "Approximate solution of Urysohn integral equations with non-smooth kernels." J. Integral Equations Applications 28 (2) 221 - 261, 2016.


Published: 2016
First available in Project Euclid: 1 July 2016

zbMATH: 1347.45004
MathSciNet: MR3518484
Digital Object Identifier: 10.1216/JIE-2016-28-2-221

Primary: 45L10 , 65J15 , 65R20

Keywords: collocation method , Gauss points , Urysohn integral operator

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.28 • No. 2 • 2016
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