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Reproducing kernel-based piecewise methods for efficiently solving oscillatory systems of second-order initial value problems

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Abstract

In this work, we will propose and analyse a novel reproducing kernel function-based piecewise approach for solving oscillatory systems of second-order initial value problems (IVPs). Also, the approach can be used to effectively solve wave equations with oscillatory solutions via the space semi-discretisation strategy. Five numerical experiments are implemented to illustrate the remarkable accuracy and efficiency of the present approach.

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References

  1. Ahmad, S.Z., Ismail, F., Senu, N., Suleiman, M.: Zero-dissipative phase-fitted hybrid methods for solving oscillatory second order ordinary differential equations. Appl. Math. Comput. 219, 10096–10104 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Aronszajn, N.: Theory of reproducing kernel. Trans. A.M.S. 168, 1–50 (1950)

    MATH  Google Scholar 

  3. Brugnano, L., Montijano, J.I., Rández, L.: On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems. Numer. Algor. 81, 345–376 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Geng, F.Z., Cui, M.G.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167–1181 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Geng, F.Z., Wu, X.: Reproducing kernel function-based Filon and Levin methods for solving highly oscillatory integral. Appl. Math. Comput. 397, 125980 (2021)

    MathSciNet  MATH  Google Scholar 

  6. Chen, Z.X., Shi, L., Liu, S.Y., You, X.: Trigonometrically fitted two-derivative Runge–Kutta–Nyström methods for second-order oscillatory differential equations. Appl. Numer. Math. 142, 171–189 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Iserles, A.: Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appl. Numer. Math. 43, 145–160 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu, C., Iserles, A., Wu, X.: Symmetric and arbitrarily high-order Birkhoff-Hermite time integrators and their long-time behaviour for solving nonlinear Klein-Gordon equations. J. Comput. Phys. 356, 1–30 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, N., Jiang, W.: A numerical method for solving the time fractional Schrödinger equation. Adv. Comput. Math. 44, 1235–1248 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Z.L., Tian, T.H., Tian, H.J.: Asymptotic-numerical solvers for highly oscillatory second-order differential equations. Appl. Numer. Math. 137, 184–202 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, W.J., Wu, B.Y., Sun, J.B.: Some numerical algorithms for solving the highly oscillatory second-order initial value problems. J. Comput. Phys. 276, 235–251 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, Z.L., Zhao, H., Tian, H.J.: Modified Filon-type methods for second-order highly oscillatory systems with a time-dependent frequency matrix. Appl. Math. Lett. 139, 108540 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  13. Papadopoulos, D.F., Anastassi, Z.A., Simos, T.E.: A phase-fitted Runge–Kutta–Nyström method for the numerical solution of initial value problems with oscillating solutions. Comput. Phys. Commun. 180, 1839–1846 (2009)

    Article  MATH  Google Scholar 

  14. Wang, Y.: On nested Picard iterative integrators for highly oscillatory second-order differential equations. Numer. Algorithms 91, 1627–1651 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wendland, H.: Scattered Data Approximation. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

  16. Wu, X., Liu, K., Shi, W.: Structure-Preserving Algorithms for Oscillatory Differential Equations II. Springer, New York (2015)

    Book  MATH  Google Scholar 

  17. Zhao, J.J., Li, Y., Xu, Y.: Multiderivative extended Runge–Kutta–Nyström methods for multi-frequency oscillatory systems. Int. J. Comput. Math. 95, 231–254 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was financially supported by the National Natural Science Foundation of China (NSFC) under Grant 11201041 and 11671200, and China Postdoctoral Science Foundation under Grant 2019M651765.

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Correspondence to Fazhan Geng.

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Geng, F., Wu, X. Reproducing kernel-based piecewise methods for efficiently solving oscillatory systems of second-order initial value problems. Calcolo 60, 20 (2023). https://doi.org/10.1007/s10092-023-00516-6

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  • DOI: https://doi.org/10.1007/s10092-023-00516-6

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