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

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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 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 in iterated form of discrete Legendre spectral collocation method in norm and uniform norm. Numerical aspects are given to verify the hypothetical results.