Fractional Jacobi-Picard iteration method using Gauss-Seidel technique for solving a system of nonlinear fractional differential equations

The main objective of this study is to introduce an improvement of Picard's method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo's fractional derivative. Using the Picard's method to solve fractional differential equations is straightforward. However, dealing with the integral in each Picard's iteration becomes tough or even impossible for nonlinear problems. Thus, we propose an iterative strategy called the fractional Jacobi-Picard iteration method, which combines Picard's iteration method with the shifted Jacobi polynomial. The computation of the fractional integrals of the shifted Jacobi polynomials is easily achieved at each step by utilizing properties of the fractional integral and shifted Jacobi polynomial. Furthermore, this approach not only transforms the system of equations into a reversible form but also solves it using the Gauss-Seidel technique. The convergence analysis of the method has been carefully performed. We performed detailed numerical simulations to show how well our method performs compared to other methods. Our results demonstrate the effectiveness and accuracy of our approach, especially in handling problems with non-smooth solutions.