A novel generalized symmetric spectral Galerkin numerical approach for solving fractional differential equations with singular kernel

Polynomial based numerical techniques usually provide the best choice for approximating the solution of fractional differential equations (FDEs). The choice of the basis at which the solution is expanded might affect the results significantly. However, there is no general approach to determine which basis will perform better with a particular problem. The aim of this paper is to develop a novel generalized symmetric orthogonal basis which has not been discussed in the context of numerical analysis before to establish a general numerical treatment for the FDEs with a singular kernel. The operational matrix with four free parameters was derived for the left-sided Caputo fractional operator in order to transform the FDEs into the corresponding algebraic system with the aid of spectral Galerkin method. Several families of the existing polynomials can be obtained as a special case from the new basis beside other new families generated according to the value of the free parameters. Consequently, the operational matrix in terms of these families was derived as a special case from the generalized one up to a coefficient diagonal matrix. Furthermore, different properties relevant to the new generalized basis were derived and the error associated with function approximation by the new basis was performed based on the generalized Taylor's formula.