A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations

Abstract

The centered difference discretization of the spatial fractional coupled nonlinear Schrödinger equations obtains a discretized linear system whose coefficient matrix is the sum of a real diagonal matrix D and a complex symmetric Toeplitz matrix T which is just the symmetric real Toeplitz T plus an imaginary identity matrix iI. In this study, we present a medium-shifted splitting iteration method to solve the discretized linear system, in which the fast algorithm can be utilized to solve the Toeplitz linear system. Theoretical analysis shows that the new iteration method is convergent. Moreover, the new splitting iteration method naturally leads to a preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are tighter than those of the original coefficient matrix A. Finally, compared with the other algorithms by numerical experiments, the new method is more effective.

Iterative positive solutions to a coupled fractional differential system with the multistrip and multipoint mixed boundary conditions

“…Many researchers established the existence of solutions by the class methods [22][23][24][25]. However, to the best of our knowledge, only few papers applied the monotone iterative technique to discuss the boundary value problem of coupled fractional differential systems [26][27][28]. To get more extensive results, different from the existing literature, we consider a generalized model that includes the nonlinear terms of the system depending on the lower fractional-order derivatives of the unknown functions and the boundary conditions involving a combination of the multistrip fractional integral and linear multipoint values of the unknown functions in [0,1].…”

Solvability for boundary value problem of the general Schrödinger equation with general superlinear nonlinearity

“…Rybalko [28] studied an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. Wen and Zhao [39] presented a medium-shifted splitting iteration method to solve the discretized linear system, in which the fast algorithm can be utilized to solve the Toeplitz linear system. Chen et al [6] investigated the existence of nontrivial solutions and multiple solutions for nonlinear Schrödinger equations with unbounded potentials.…”