Discrete Legendre spectral methods for Hammerstein type weakly singular nonlinear Fredholm integral equations

Abstract

In this article, we study the discrete version of Legendre spectral and iterated Legendre spectral techniques to solve the second kind Hammerstein type weakly singular integral equations. To obtain the convergence analysis, we use the appropriate numerical quadrature rule and obtain the order $\mathcal{O}(n^{-r+1})$ in discrete Legendre spectral method. If the quadrature rule is minimal, i.e. the number of quadrature nodes and the dimension of the approximating subspace are same, then the optimal rate is obtained $\mathcal{O}(n^{-r})$ in iterated form of discrete Legendre spectral collocation method in $L^2$ norm and uniform norm. Numerical aspects are given to verify the hypothetical results.

Keywords:

• Hammerstein-type integral equations
• Galerkin method
• weakly singular kernels
• Legendre polynomial
• convergence costs

2010 Mathematics Subject Classifications:

• 45B05
• 45G10
• 65R20

Acknowledgments

The research work of Gnaneshwar Nelakanti was supported by the National Board for Higher Mathematics, India, research project: No02011/6/2019NBHM(R.P)/R & D II /1236 dated 28/1/2019.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research work of Gnaneshwar Nelakanti was supported by the National Board for Higher Mathematics, India, research project: No. 02011/6/2019NBHM(R.P)/R & D II /1236 dated 28/1/2019.