Abstract
This research describes an efficient numerical method based on Wendland’s compactly supported functions to simulate the time-space fractional coupled nonlinear Schrödinger (TSFCNLS) equations. Here, the time and space fractional derivatives are considered in terms of Caputo and Conformable derivatives, respectively. The present numerical discussion is based on the following ways: we first approximate the Caputo fractional derivative of the proposed equation by a scheme order O(∆t2−α), 0 < α < 1 and then the Crank-Nicolson scheme is employed in the mentioned equation to discretize the equations. Second, applying a linear difference scheme to avoid solving nonlinear systems. In this way, we have a linear, suitable calculation scheme. Then, the conformable fractional derivatives of the Wendland’s compactly supported functions are established for the scheme. The stability analysis of the suggested scheme is also examined in a similar way to the classic Von-Neumann technique for the governing equations. The efficiency and accuracy of the present method are verified by solving two examples.
Highlights
Our main intention of this current investigation is to simulate the time-space fractional coupled nonlinear Schrodinger (TSFCNLS) equations ∂αu i ∂tα ∂βu + γ ∂|x|β+ ρ(|u|2 + η|v|2)u = 0,(x, t) ∈ [a, b] × [0, T ], (1.1) ∂αv i ∂tα ∂βv + γ ∂|x|β+ ρ(η|u|2 + |v|2)v = 0, (1.2)
TSFCNLS equations are not solved by using the collocation method based on Wendland’s compactly supported functions
We have proposed a method based on Wendland’s compactly supported functions to solve numerically the TSFCNLS equations
Summary
Our main intention of this current investigation is to simulate the TSFCNLS equations. In case α = 1 and β = 2, this system represents classical coupled nonlinear Schrodinger (CNLS) equations. When η = 0, the system (1.1)–(1.2) reduces to single time and space fractional nonlinear Schrodinger (TSFNLS) equation. Some physical applications of the fractional Schrodinger equations with time-space fractional derivatives are studied by the authors [8, 30]. The finite difference method is constructed for time-space fractional Schrodinger equations by Liu, Zheng and. Another study is done by [22] They used a linearized Crank-Nicolson method to obtain the numerical solution of the time-space fractional NLS equations. The papers [21, 29] used a conservative difference method to obtain the numerical solutions of space fractional CNLS equations.
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