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Abstract In quantum physics, fractional Schrödinger equation is of particular interest in the research of particles on stochastic fields modeled by the Lévy processes, which was derived by extending the Feynman path integral over the Brownian paths to a path integral over the trajectories of Lévy fights. In this work, a fully discrete finite element method (FEM) is developed for the Riesz space-fractional coupled nonlinear Schrödinger equations (CNLS), conjectured with a linearized Crank–Nicolson discretization. The error estimate and mass conservative property are discussed. It is showed that the proposed method is decoupled and convergent with optimal orders in L2 $L^{2}$-sense. Numerical examples are performed to support our theoretical results.

Abstract In this article, we propose an exponential B-spline approach to obtain approximate solutions for the fractional sub-diffusion equation of Caputo type. The presented method is established via a uniform nodal collocation strategy by using an exponential B-spline based interpolation in conjunction with an effective finite difference scheme in time. The unique solvability is rigorously proved. The unconditional stability is well illustrated via a procedure closely resembling the classic von Neumann technique. A series of numerical examples are carried out, and by contrast to other algorithms available in the open literature, numerical results confirm the validity and superiority of our method.