Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations

TL;DR

This paper introduces a new collocation method to detect blow-up solutions in nonlinear Volterra integral equations, providing numerical estimates and analyzing the relationship between approximate and exact blow-up behaviors.

Contribution

It presents the concept of blow-up collocation solutions and analyzes their properties, advancing numerical techniques for studying blow-up phenomena in integral equations.

Findings

Numerical estimates of blow-up times using collocation methods.

Relationships between blow-up conditions for collocation and exact solutions.

Application of the method to specific nonlinear Volterra equations.

Abstract

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this, we introduce the concept of "blow-up collocation solution" and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we discuss the relationships between necessary conditions for blow-up of collocation solutions and exact solutions.