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Solving time-dependent differential equations by multiquadric trigonometric quasi-interpolation

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  • Gao, Wenwu
  • Wu, Zongmin

Abstract

Multiquadric (MQ) quasi-interpolation is a popular method for the numerical solution of differential equations. However, MQ quasi-interpolation is not well suited for the equations with periodic solutions. This is mainly due to the fact that its kernel (the MQ function) is not a periodic function. A reasonable way of overcoming the difficulty is to use a quasi-interpolant whose kernel itself is also periodic in these cases. The paper constructs such a quasi-interpolant. Error estimates of the quasi-interpolant are also provided. The quasi-interpolant possesses many fair properties of the MQ quasi-interpolant (i.e., simplicity, efficiency, stability, etc). Moreover, it is more suitable (than the MQ quasi-interpolant) for periodic problems since the quasi-interpolant as well as its derivatives are periodic. Examples of solving both linear and nonlinear partial differential equations (whose solutions are periodic) by the quasi-interpolant and the MQ quasi-interpolant are compared at the end of the paper. Numerical results show that the quasi-interpolant outperforms the MQ quasi-interpolant for periodic problems.

Suggested Citation

  • Gao, Wenwu & Wu, Zongmin, 2015. "Solving time-dependent differential equations by multiquadric trigonometric quasi-interpolation," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 377-386.
    • Handle: RePEc:eee:apmaco:v:253:y:2015:i:c:p:377-386
    DOI: 10.1016/j.amc.2014.12.008
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    Cited by:

    1. Shenggang Zhang & Chungang Zhu & Qinjiao Gao, 2019. "Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation," Mathematics, MDPI, vol. 7(8), pages 1-15, August.
    2. Wu, Hui-Yuan & Duan, Yong, 2016. "Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis–Procesi equation," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 83-92.

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