Numerical solution of certain Cauchy singular integral equations using a collocation scheme

• Published in: Advances in difference equations Vol. 2020; no. 1; pp. 1 - 15
• Main Author: Seifi, Ali
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 29.09.2020; SpringerOpen
• Subjects: Analysis; Applications; Bessel polynomial; Cauchy singular integral equation; Collocation scheme; Difference and Functional Equations; Functional Analysis; Mathematics; Mathematics and Statistics; Methods; Ordinary Differential Equations; Partial Differential Equations; Topics in Special Functions and q-Special Functions: Theory; Bessel polynomial; Collocation scheme; Cauchy singular integral equation
• ISSN: 1687-1847; 1687-1847
• DOI: 10.1186/s13662-020-02996-0

The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.