Jacobi collocation method and smoothing transformation for numerical solution of neutral nonlinear weakly singular Fredholm integro-differential equations

Abstract

In this work, we consider Jacobi collocation method for the numerical solution of neutral nonlinear weakly singular Fredholm integro-differential equations. We propose conditions that under them the equation has a unique solution. Since the second derivative of the solution is singular at the end points of the interval, we have to use a smoothing transformation to improve the smoothness of the solution. Then, we use Jacobi points systems to collocate the solution and also Gauss quadrature rules to approximate the integrals. For the error analysis, we define integral and quadrature operators and using a result for the interpolation error of functions with limited regularity, we derive an error bound for the numerical solution. The proposed method is applied on some numerical examples and the errors are reported for different parameters of the smoothing transformation.