Smoothing transformation for numerical solution of nonlinear weakly singular Volterra integral equations using quasilinearization and product integration methods

Abstract

We consider extension of quasilinearization technique and method of lower and upper solutions for nonlinear weakly singular Volterra integral equations. We obtain linear weakly singular integral equations and discuss on their existence, uniqueness and regularity properties of a solution. On the other hand, we show that the solutions of these linear equations are quadratically convergent to the solution of nonlinear equation. Using smoothing transformation we regularize the linear equations and then to approximate their solutions we apply global product integration method. Because of employing quasilinearization technique and smoothing transformation we yield a sequence of small size linear algebraic systems in the discretization. Error analysis shows the error bound has two parts of quasilinearization error which is quadratically convergent and product integration error where by using smoothing transformation its convergence order is improved. The numerical results obtained from different numerical examples are in agreement with the theoretical results and comparison with the other methods confirms the efficiency of the proposed method.