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Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

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Abstract

The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artificial boundary method, we first derive the exact artificial nonreflecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458–3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonreflecting boundary condition and leads us to reformulate the original infinite discrete system into an equivalent finite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are finally provided to demonstrate the effectiveness of our approach.

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Acknowledgements

The second author thanks professor Qiang Du’s useful suggestions. Jiwei Zhang is partially supported by the National Natural Science Foundation of China under Grant No. 11771035 and the NSAF U1530401, and the Natural Science Foundation of Hubei Province No. 2019CFA007, and Xiangtan University 2018ICIP01. Chunxiong Zheng is partially supported by Natural Science Foundation of Xinjiang Autonomous Region under No. 2019D01C026, and the National Natural Science Foundation of China under Grant Nos. 11771248 and 91630205. Jiwei Zhang thanks for the support and hospitality of Beijing Computational Science Research Center during his visiting period.

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Yan, Y., Zhang, J. & Zheng, C. Numerical Computations of Nonlocal Schrödinger Equations on the Real Line. Commun. Appl. Math. Comput. 2, 241–260 (2020). https://doi.org/10.1007/s42967-019-00052-7

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  • DOI: https://doi.org/10.1007/s42967-019-00052-7

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