A numerical algorithm for solving index-1 weakly singular integral-algebraic equations with non-smooth solutions

Abstract

This paper aims to present a product integration approach based on the first kind of Chebyshev polynomials to solve the weakly singular integral-algebraic equations of index-1. These are mixed systems of weakly singular Volterra integral equations of the second and first kind. The solutions of these equations have a singularity at the lower bound of the domain of integration. To deal with this non-smooth behavior of solutions, we apply the suitable transformations to get a new system with regular solutions. We use the differentiation index in the theoretical part. One of the advantages of selecting the weights of the presented numerical method is that when we determine the error bounds, some expressions will be equal to zero speeding up the process of achieving the error bounds. Finally, by providing some numerical examples, we will observe that the numerical results confirm the validity of convergence analysis.

Keywords:

• Semi-explicit
• non-smooth behavior
• integral-algebraic equations
• differentiation index
• weakly singular

Mathematics Subject Classifications 2020:

• 45F15
• 45Dxx

Disclosure statement

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

Additional information

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.