Two reliable methods for solving the Volterra integral equation with a weakly singular kernel

Two reliable methods, namely the Adomian decomposition method (ADM) and the variational iteration method (VIM), are used for solving the Volterra integral equation with a weakly singular kernel in the reproducing kernel space. Both methods provide convergent series solutions for this equation. The ADM method gives a sequence of components of the solution, which composes a sequence of approximations, whereas the VIM more directly provides a sequence of approximations; both exhibit high accuracy. Four numerical examples are examined to confirm the validity and the power of these two methods.