A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations

Abstract

Abstract(#br)In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $$(x-s)^{-\mu },0<\mu <1$$ ( x - s ) - μ , 0 < μ < 1 . First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both $$L^{\infty }$$ L ∞ - and weighted $$L^{2}$$ L 2 -norms. The main novelty of the paper is that the..