# A M\"untz-Collocation spectral method for weakly singular volterra integral equations

## TL;DR

This paper introduces a fractional Jacobi-collocation spectral method for solving weakly singular Volterra integral equations, achieving exponential convergence for smooth solutions after a variable transformation, with demonstrated numerical efficiency.

## Contribution

The paper develops a new fractional Jacobi polynomial family and constructs an efficient spectral method with proven exponential convergence for weakly singular VIEs.

## Key findings

- Method achieves exponential convergence for smooth solutions.
- Numerical examples confirm high efficiency and accuracy.
- New fractional Jacobi polynomials facilitate spectral approximation.

## Abstract

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 $(x-s)^{-\mu},0<\mu<1$. 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 $L^{\infty}$- and weighted $L^{2}$-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change $x\rightarrow x^{1/\lambda}$ for a suitable real number $\lambda$. Finally a series of numerical examples are presented to demonstrate the efficiency of the method.