Chenkuan Li

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.

1 Introduction and Preliminaries The Riemann–Liouville fractional integral I β $$ {I}^{\beta } $$ of order β ∈ R + $$ \beta \in {R}^{+} $$ is defined for the function ζ ( x ) $$ \zeta (x) $$ as ( I β ζ ) ( x ) = 1 Γ ( β ) ∫ 0 x ( x − τ ) β − 1 ζ ( τ ) d τ , x ∈ [ 0 , T ] , T > 0 . $$ \left({I}^{\beta}\zeta \right)(x)=\frac{1}{\Gamma \left(\beta \right)}{\int}_0^x{\left(x-\tau \right)}^{\beta -1}\zeta \left(\tau \right) d\tau, \kern1em x\in \left[0,T\right],\kern3.0235pt T>0.