New numerical analysis of Riemann-Liouville time-fractional Schrödinger with power, exponential decay, and Mittag-Leffler laws

Badr Saad T. Alkahtani, Ilknur Koca, Abdon Atangana
2017 Journal of Nonlinear Science and its Applications  
The mathematical equation that describes how the quantum state of a quantum system changes during time was considered within the framework of fractional differentiation with three different derivatives in Riemann-Liouville sense. The fractional derivatives used in this work are constructed based on power, exponential decay, and Mittag-Leffler law. A new numerical scheme for fractional derivative in Riemann-Liouville sense is presented and used to solve numerically the Schrödinger equation. The
more » ... nger equation. The stability analysis of each model is presented in detail. 4232 law kernel was suggested by Caputo and Fabrizio [3, 8, 9, 12] . However, there is an established rule in the scope of fractional differentiation that states that any fractional derivative must have a non-local kernel. With this established rule, it appears that the Caputo-Fabrizio derivative is not a fractional operator however it is very useful in describing real world problems that follow the exponential decay law. To solve the problem of singularity, non-locality of the kernel and also to have added the more generalized decay law, Abdon Atangana and Dumitru Baleanu suggested a fractional derivative with the generalized Mittag-Leffler function as kernel [1, 2, 4, 6] . The definition of fractional derivative based on the Riemann-Liouville approach although very useful but it numerical approximation has not gain attention in the field of fractional differentiation. Recently Atangana and Gomez suggested the numerical approximation of fractional derivative from power law to the generalized Mittag-Leffler function. In this paper, we solve numerically the Schrödinger equation with power law, exponential decay law and the generalized Mittag-Leffler law as kernels of time fractional derivative. Fractional order derivatives in Riemann-Liouville sense In this section, we present the fractional order definitions with Riemann-Liouville senses. Definition 2.1. Let f be a function not necessarily differentiable, and α be a real number such that 0 α 1, then the derivative with α order with power law is given as [13] RL D α t [f(t)] =
doi:10.22436/jnsa.010.08.18 fatcat:vhwpxxveefcyhohonp2rfmwjti