Error estimates of generalized spectral iterative methods with accurate convergence rates for solving systems of fractional two - point boundary value problems

Abstract

The main purpose of this study is to provide an efficient spectral iterative method based on fractional interpolants for solving a class of linear and nonlinear fractional two - point boundary value problems involving left and right fractional derivatives. In order to achieve this goal, by composition of iterative method and spectral method, in each step, the unknown solution of system is expanded by using two classes of generalized Jacobi functions to obtain numerically coefficients. For this system of fractional differential equations on the interval [−1,1], the singularities of the solution are considered of types (1+t)β and (1−x)α, where 0 < β, α < 1 are left and right singularities indexes. Then, these types of singularities can be well resolved with the mentioned functions. A rigours convergence analysis of the proposed method with accurate spectral rate of convergence is extensively discussed in L2−norm. Finally, some numerical results are given to demonstrate the effectiveness and applicability of the proposed method and accuracy of the presented convergence rates.