Analysis of Polynomial Collocation and Uniformly Spaced Quadrature Methods for Second Kind Linear Fredholm Integral Equations – A Comparison

Muhammad Mujtaba Shaikh^{1, 2,}

¹Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan

²Supply Chain and Operations Management Research Group, Mehran University of Engineering and Technology, Jamshoro, Pakistan

Cite this paper:
Muhammad Mujtaba Shaikh. Analysis of Polynomial Collocation and Uniformly Spaced Quadrature Methods for Second Kind Linear Fredholm Integral Equations – A Comparison. Turkish Journal of Analysis and Number Theory. 2019; 7(4):91-97. doi: 10.12691/tjant-7-4-1

Abstract

This work aims to compare the polynomial collocation method against the uniformly spaced quadrature methods (Trapezoidal, Simpson’s and Weddle’s rules) to solve the non-homogeneous linear Fredholm integral equations of second kind with non-singular kernel. Nystrom’s interpolation technique is used to attain closed form approximations of the solutions obtained by quadrature methods. The formulation and implementation of the methods along with application on various test problems are presented for comparison. The results obtained, in general, highlight some cases where polynomial collocation yields ill-conditioned systems after discretization with same basis set, and reflects the suitability of quadrature rules. The main focus has been to extract exact closed form solutions by the used methods which can be used as replacement of analytical solutions.

Keywords:
Fredholm integral equation polynomial collocation quadrature rules Nystrom’s interpolation

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