Space-time Müntz spectral collocation approach for parabolic Volterra integro-differential equations with a singular kernel

We consider a type of Volterra integro-differential equations of the parabolic type that arise naturally in the study of heat flow in materials with memory. We present a simple and accurate numerical method for problems with a weakly singular kernel subject to an initial condition and given boundary conditions. In this method, both the space and time discretizations are based on the M¨untz-Legendre collocation method that converts the problem to a system of algebraic equations. For numerical stability purposes, the M¨untz-Legendre polynomials and their partial derivatives are stated in terms of Jacobi polynomials. Moreover, to deal with the weakly singular integral term of the problem, two efficient schemes based on the integration by parts and nonclassical Gaussian quadrature are derived. Comparisons between the two proposed schemes and other methods in the literature are made to demonstrate the efficiency, convergence and superiority of our method in the space and time directions.