Numerical algorithm to solve a coupled system of fractional order using a novel reproducing kernel method

In this paper, a coupled system of fractional differential equations along with integral boundary conditions is discussed by means of the iterative reproducing kernel algorithm. Towards this end, a recently advanced analytical approach is proposed to obtain approximate solutions of nonclassical-types boundary value problems of fractional derivatives in Caputo sense. This approach optimizes approximate solutions based on the Gram-Schmidt process on Sobolev spaces that execute to generate Fourier expansion within a fast convergence rate, whereby the constructed kernel function fulfills homogeneous integral boundary conditions. Moreover, the solution is presented in the form of a fractional series over the entire Hilbert spaces without unwarranted assumptions on the considered models. The validity of the present algorithm is illustrated by expounding and testing two numerical examples. The achieved results indicate that the proposed algorithm is systematic, feasibility, stability, and convenient for dealing with other fractional systems emerging in the physical, technology and engineering.