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An Effective Iteration Method for a Class of Nonlinear Singular Perturbed Problems

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Abstract

In this paper we investigate a class of singular perturbed differential equations with initial conditions. Based on the equations cannot solved easily by the traditional methods, we give a simple but effective method for finding their approximate periodic solutions firstly. That is, we introduce an iterative technique to the singular perturbed nonlinear system and then adopt the classical perturbed technique to give the approximate solutions. Secondly, some typical examples given to illustrate the methods is a useful tool in solving the nonlinear singular perturbed systems. At last, we present a windowing technique for the system. The convergence processes can be speeded up when we take the windowing technique. These methods can be easily extended to other nonlinear systems and found widely applicable in engineering.

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References

  1. Wollkind, D.J.: Singular perturbation techniques: a comparison of the method of matched asymptotic expansions with that of multiple scles. SIAM Rev. 19(3), 502–516 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Nafeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)

    Google Scholar 

  3. Holmes, M.H.: Introduction to Perturbation Methods. World Scientific Publishing, Singapore (1999)

    Google Scholar 

  4. Boglaev, I.: A uniform monotone alternating direction scheme for nonlinear singularly perturbed parabolic problems. J. Comput. Appl. Math. 272, 148–161 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Xie, F.: On a class of singular boundary value problems with singular perturbation. J. Differ. Equ. 252(3), 2370–2387 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Franz, S., Kopteva, N.: Green’s function estimates for a singularly perturbed convection–diffusion problem. J. Differ. Equ. 252(2), 1521–1545 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Meyer, K.R., Palacian, J.F., Yanguas, P.: The elusive Liapunov periodic solutions. Qual. Theor. Dyn. Syst. 14(2), 381–401 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ducrot, A., Langlais, M.: A singular reaction–diffusion system modelling prey–predator interactions: invasion and co-extinction waves. J. Differ. Equ. 253(2), 502–532 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boglaev, I.: Uniform quadratic convergence of monotone iterates for nonlinear singularly perturbed parabolic problems. Numer. Algorithms 64(4), 607–631 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dumortier, F.: Slow divergence integral and balanced canard solutions. Qual. Theor. Dyn. Syst. 10(1), 65–85 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mickens, R.E.: Iteration procedure for determining approximate solutions to non-linear oscillator equations. J. Sound Vib. 116(1), 185–187 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Y.M., Liu, J.K.: A modied Mickens iteration procedure for nonlinear oscillators. SIAM J. Numer. Anal. 314, 465–473 (2008)

    Google Scholar 

  13. Lim, C.W., Wu, B.S.: A new analytical approach to the Duffing-harmonic oscillator. Phys. Lett. A. 311(4–5), 365–373 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Marinca, V., Herisanu, N.: An optimal iteration method for strongly nonlinear oscillators. J. Appl. Math. 906341, 1–11 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Song, B., Jiang, Y.L.: Analysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems. Numer. Algorithms 67(3), 599–622 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jiang, Y.L.: Windowing waveform relaxation of initial value problems. Acta. Math. Appl. Sin. E 22(4), 575–588 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leimkuhler, B.: Estimating waveform relaxation convergence. SIAM J. Sci. Comput. 14(4), 872–889 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sand, J., Burrage, K.: A Jacobi waveform relaxation method for ODEs. SIAM J. Sci. Comput. 20(2), 534–552 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, Y.L., Chen, R.M.M., Wing, O.: Improving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation. IEEE Trans. Circuits Syst I 48(6), 769–780 (2001)

    Article  MATH  Google Scholar 

  20. Lumsdaine, A., Reichelt, M.W., Squyres, J.M., White, J.K.: Accelerated waveform methods for parallel transient simulation of semiconductor devices. IEEE Trans. Comput. Aided Des. 15(7), 716–726 (1996)

    Article  Google Scholar 

  21. Jiang, Y.L.: A general approach to waveform relaxation solutions of differential-algebraic equations: the continuous-time and discrete-time cases. IEEE Trans. Circuits Syst I 51(9), 1770–1780 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jiang, Y.L.: Periodic waveform relaxation solutions of nonlinear dynamic equtions. Appl. Math. Comput. 135(2–3), 219–226 (2003)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (11401420), the Natural Science Foundation of Shanxi (201601D102002), Special Foundation of Taiyuan University of Technology (2015MS033) and the Talent Introduction Research Fund of Taiyuan University of Technology (tyut-rc201317a).

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Authors and Affiliations

  1. College of Mathematics, Taiyuan University of Technology, No. 79, West Yingze Street Taiyuan, Taiyuan, 030024, Shanxi, China

    Xiaoyun Wang, Yuyang Pan & Liang Bai

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  1. Xiaoyun Wang
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  2. Yuyang Pan
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  3. Liang Bai
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Correspondence to Xiaoyun Wang.

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Wang, X., Pan, Y. & Bai, L. An Effective Iteration Method for a Class of Nonlinear Singular Perturbed Problems. Qual. Theory Dyn. Syst. 17, 155–175 (2018). https://doi.org/10.1007/s12346-017-0240-5

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  • DOI: https://doi.org/10.1007/s12346-017-0240-5

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