An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions

This paper is concerned with the numerical solution of Volterra integral equations of the third kind with non-smooth solutions based on the recursive approach of the spectral Tau method. To this end, a new set of the fractional version of canonical basis polynomials (called FC-polynomials) is introduced. The approximate polynomial solution (called Tau-solution) is expressed in terms of FC-polynomials. The fractional structure of Tau-solution allows recovering the standard degree of accuracy of spectral methods even in the case of non-smooth solutions. The convergence analysis of the method is carried out. The obtained numerical results show the accuracy and efficiency of the method compared to other existing methods.