A priori mesh grading in collocation solution of noncompact Volterra integral equations with diagonal singularity

Abstract

This work is devoted to some theoretical results concerning the collocation method on a graded mesh for a class of noncompact cordial Volterra integral equations whose kernels possess diagonal singularity depending on a real parameter $\nu \in (0,1)$. The existence and uniqueness with regularity properties of the solution are analysed. Our main focus is to choose the mesh points in such a way that the highest possible order of convergence is attained. To do this, we consider the spline collocation method with a graded mesh by an optimal scaling parameter which gives higher order of convergence. Due to the instability of the error behaviours for large values of grading exponent, choosing the optimal value of $r=r(\nu )$ is a crucial issue from a numerical point of view. We will provide a sufficient condition on the grading exponent which controls the grading and characterizes the density of the mesh points. The global convergence as well as the optimal superconvergence results of the solutions on such a graded mesh are also investigated. Finally, some results of numerical experiments are reported which verify the validity of the theoretical results.

Keywords:

• Volterra integral equation
• diagonal singularity
• noncompact kernel
• collocation method
• graded mesh

2020 AMS Subject Classifications:

• 65R20
• 45A05
• 45P05

Disclosure statement

No potential conflict of interest was reported by the author(s).