Stability and error analysis of the reproducing kernel Hilbert space method for the solution of weakly singular Volterra integral equation on graded mesh

In this article, we approximate the solution of the weakly singular Volterra integral equation of the second kind using the reproducing kernel Hilbert space (RKHS) method. This method does not require any background mesh and can easily be implemented. Since the solution of the second kind weakly singular Volterra integral equation has unbounded derivative at the left end point of the interval of the integral equation domain, RKHS method has poor convergence rate on the conventional uniform mesh. Consequently, the graded mesh is proposed. Using error analysis, we show the RKHS method has better convergence rate on the graded mesh than the uniform mesh. Numerical examples are given to confirm the error analysis results. Regularization of the solution is an alternative approach to improve the efficiency of the RKHS method. In this regard, an smooth transformation is used to regularization and obtained numerical results are compared with other methods.