A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions

Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval (0, T ). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both L 2 - and L^∞ - norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper.