A Müntz spectral method for solving fractional Fredholm integro–differential equations with convergence analysis

The solutions of fractional order equations might involve certain fractional-power terms that classical orthogonal polynomials connot match. Consequently, the advancement of effective numerical methods using generalized orthogonal polynomials such as fractional Jacobi, Müntz and fractional Chelyshkov functions enhances the precision of approximate solutions. This paper proposes a novel Müntz–Legendre spectral approach for a class of fractional Fredholm integro–differential equations with Caputo or Caputo–Fabrizio (CF) derivative. We first construct a matrix method that transforms the given linear problem to a system of linear algebraic equations. Then, we give a comprehensive convergence analysis of the proposed method. As opposed to the Caputo definition, the derivative of CF has no singularity at the end point, so it is expected that it is more convenient for numerical studies. Nonetheless, we propose a new approach to deal with the singularity in the definition of the Caputo derivative, efficiently. Some numerical examples are given and comparisons with other existing methods are provided to demonstrate the efficiency and accuracy of the proposed method. The extension of the proposed method to nonlinear problems via the linearization technique is also illustrated in an example.