Numerical Solutions of Weakly Singular Nonlinear Volterra Integral Equations With Smooth and Nonsmooth Solutions Over Large Time Intervals

ABSTRACT This paper proposes a spectral collocation method for solving weakly singular nonlinear Volterra integral equations over large time intervals, addressing both smooth and nonsmooth solutions. For smooth solutions, classical scaled Laguerre polynomials (SLPs) are employed. For nonsmooth solutions, a new set of generalized Laguerre functions (GLFs) is introduced. To efficiently approximate weakly singular integrals, the interval is transformed into a half‐line using a suitable mapping. The Gauss points of the GLFs are then used as the collocation points to approximate the resulting integral. Also, the convergence of the proposed method is analyzed in the weighted ‐norm. Finally, the accuracy and reliability of the new approach are demonstrated through several examples.