A Gegenbauer polynomial-based numerical technique for a singular integral equation of order four and its application to a crack problem.

Strongly singular integral equations of order four have applications in fracture mechanics, and Gegenbauer polynomials have never been used to solve these equations. This motivated us to develop a Gegenbauer polynomial-based Galerkin method to solve a singular integral equation of order four. We first prove the problem's well-posedness. Then, we show the theoretical convergence of the numerical scheme and derive the rate of convergence and the error estimates. We validate the theoretical error estimates numerically in test examples. We implement the proposed method to a crack problem and compare it with existing results in the literature. • A residual-based Galerkin's method using Gegenbauer polynomials has been developed. • The well-posedness of a strong singular integral equation has been shown using the operator theory. • The error bounds, convergence as well as the rate of convergence has been theoretically derived. • Comparison between our proposed method and the existing collocation method has been shown numerically. [ABSTRACT FROM AUTHOR]