An easy-to-implement recursive fractional spectral-Galerkin method for multi-term weakly singular Volterra integral equations with non-smooth solutions

Multi-term weakly singular Volterra integral equations arise in numerous scientific and engineering applications modeled by fractional differential equations. Numerical analysis of these types of integral equations with non-smooth solutions is one of the open problems due to singularities at the initial time leading to the destruction of the accuracy of the approximate solutions. The spectral approximations based on classical orthogonal polynomials have a low convergence rate compared with the exact non-smooth solutions. We theoretically investigate the regularity of the solution of multi-term weakly singular Volterra integral equations, which ensures the non-smooth property of the solutions. This non-smooth representation is significant in constructing highly accurate numerical schemes based on fractional-order approximate solutions. The advantage of the proposed method is that there is no need to solve an algebraic system of equations, as the unknown coefficients can be specified by a simple recursive algorithm. This makes the numerical method for solving the problem efficient and of low computational cost. The L_∞ -convergence analysis of the proposed method is discussed. Some numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed method.