Electron. J. Differential Equations,
Vol. 2017 (2017), No. 166, pp. 1-18.
Finite element method for time-space-fractional Schrodinger equation
Xiaogang Zhu, Zhanbin Yuan, Jungang Wang, Yufeng Nie, Zongze Yang
Abstract:
In this article, we develop a fully discrete finite element method for
the nonlinear Schrodinger equation (NLS) with time- and space-fractional
derivatives. The time-fractional derivative is described in Caputo's sense
and the space-fractional derivative in Riesz's sense.
Its stability is well derived; the convergent estimate is discussed by an
orthogonal operator. We also extend the method to the two-dimensional
time-space-fractional NLS and to avoid the iterative solvers at each time
step, a linearized scheme is further conducted. Several numerical examples
are implemented finally, which confirm the theoretical results as well as
illustrate the accuracy of our methods.
Submitted January 30, 2016. Published July 5, 2017.
Math Subject Classifications: 35R11, 65M60, 65M12.
Key Words: Time-space-fractional NLS; finite element method; convergence.
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Xiaogang Zhu
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an 710129, China
email: zhuxg590@yeah.net
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Zhanbin Yuan
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an 710129, China
email: yzzzb@nwpu.edu.cn
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Jungang Wang
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an 710129, China
email: wangjungang@nwpu.edu.cn
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Yufeng Nie
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an 710129, China
email: yfnie@nwpu.edu.cn
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Zongze Yang
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an 710129, China
email: yangzongze@gmail.com
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