A regularization–discretization strategy for weakly singular Fredholm integral equations: a Tikhonov–Legendre approach
Open Mathematics, vol.24, no.1, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 24 Issue: 1
- Publication Date: 2026
- Doi Number: 10.1515/math-2025-0273
- Journal Name: Open Mathematics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, zbMATH, Directory of Open Access Journals, Technology Collection (ProQuest)
- Keywords: Gauss–Jacobi quadrature, ill-posed problems, Legendre polynomials, Tikhonov regularization, weakly singular integral equations
- Istanbul Gelisim University Affiliated: Yes
Abstract
This study presents a new computational technique for solving weakly singular Fredholm integral equations of the first kind, based on a regularization–discretization strategy that combines Tikhonov regularization with Legendre spectral projection methods. To incorporate the singular part of the kernel, we divide the integration into two parts and employ a straightforward variable transformation using two Jacobi weight functions. The presented numerical method effectively transforms the solution of weakly singular integral equations of the first kind into the solution of a linear system of algebraic equations. A thorough theoretical analysis of the proposed technique is presented. Finally, a series of numerical tests were conducted to demonstrate the validity and efficiency of our approach.