New generalized Jacobi–Galerkin operational matrices of derivatives: an algorithm for solving the time-fractional coupled KdV equations

The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances.