Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions

This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best candidate to overcome the drawbacks of the accuracy of the spectral methods. On the other hand, the fractional integration of the fractional polynomials functions is in the class of fractional polynomials and this is one of the main advantages of using the fractional basis functions. In this paper, an implicit spectral collocation method based on the fractional Chelyshkov basis functions is introduced. The framework of the method is to reduce the problem into a nonlinear system of equations utilizing the spectral collocation method along with the fractional operational integration matrix. The obtained algebraic system is solved using Newton's iterative method. Convergence analysis of the method is studied. The numerical examples show the efficiency of the method on the problems with smooth and non-smooth solutions in comparison with other existing methods.