A Variable Lavrent’ev Regularized Collocation Method for Auto-Convolution Volterra Integral Equations of the First Kind

In this paper, based on the generalized Lavrent’ev regularization, we proposed a piecewise polynomial collocation method for efficiently solving kernel-based auto-convolution Volterra integral equations of the first kind with noisy right-hand side. Our approach allows a variable regularization parameter selection rule, which is useful for the problem with different spatial ill-posedness. Under the standard smoothness assumptions of the exact solution, we prove the well-posedness as well as the regularization property (i.e. the stability with respect to the noise) of the generalized Lavrent’ev regularized collocation solution. Several numerical examples are provided to show the simplicity and efficiency of the proposed method.