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The shock solution to a class of nonlinear singularly perturbed boundary value problems with time delay

  • School of Mathematics and Physics, Anhui University of Technology, Maanshan 243002,China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.02.005
  • Received Date: 18 June 2019
  • Accepted Date: 24 July 2019
  • Rev Recd Date: 24 July 2019
  • Publish Date: 28 February 2020
    Abstract

  • Abstract

    The shock solution for a class of singularly perturbed time delay nonlinear problem were studied .The solution was obtained by using the matching asymptotic expansion,and the uniform validity of the shock solution of the problem was proved by the theory of differential inequalities .Finally, an example was given to verify the existence of the shock solution.

    Abstract

    The shock solution for a class of singularly perturbed time delay nonlinear problem were studied .The solution was obtained by using the matching asymptotic expansion,and the uniform validity of the shock solution of the problem was proved by the theory of differential inequalities .Finally, an example was given to verify the existence of the shock solution.
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    • References(12)
  • [1]
    O’MALLEY R E JR. Introduction to Singular Perturbation[M]. New York: Academic Press, 1974.
    [2]
    DE JAGER E M, JIANG F. The Theory of Singular Perturbation[M]. Amsterdam: North-Holland Publishing Co, 1996.
    [3]
    NAYFEH A H. Introduction for Perturbation Techniques[M]. New York: John Wiley & Sons, 1981.
    [4]
    CHANG K W, HOWES F A. Nonlinear Singular Perturbation Phenomena: Theory and Application[M]. New York: Springer Verlag, 1984.
    [5]
    BOH A.The shock location for a class of sensitive boundary value problems[J]. J of Mathematical Analysis and Applications, 1999, 235(1): 295-314.
    [6]
    MO Jiaqi. The shock solutions for a class of singularly perturbed time delay boundary value problems[J]. Journal of Anhui Normal University (Natural Science), 2013, 36(4): 314-318.
    [7]
    XU Yonghong, SHI Lanfang, MO Jiaqi. Boundary perturbed problem for reaction diffusion time delay equation with two parameters[J]. Journal of Wuhan University (Natural Sciences), 2015, 20(2): 93-96.
    [8]
    MO Jiaqi, WANG Weigang, CHEN Xianfeng, et al. The shock wave solutions for singularly perturbed time delay nonlinear boundary value problems with two parameters[J]. Mathematica Applicata, 2014, 27(3): 470-475.
    [9]
    YAO Jingsun, MO Jiaqi. Singularly perturbed solution to semilinear higher order reaction diffusion equations with two parameters[J]. Annals of Differential Equations, 2009,25(1): 91-96.
    [10]
    FENG Yihu, MO Jiaqi. The shock asympototic solution for nonlinear elliptic equation with two parameters[J]. Mathematica Applicata, 2015, 28(3): 579-585.
    [11]
    朱红宝.一类非线性奇摄动时滞边值问题的激波解[J].中国科学技术大学学报,2018,48(5):357-360.
    ZHU Hongbao.The shock solution to a class of singularly perturbed time delay nonlinear boundary value problem[J].Journal of University of Science and Technology of China,2018,48(5):357-360.
    [12]
    TANG Rongrong. Solution with shock-boundary layer and shock-interior layer to a class of nonlinear problems[J]. Annals of Differential Equations, 2012, 28(1): 87-92.)
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    [1]
    O’MALLEY R E JR. Introduction to Singular Perturbation[M]. New York: Academic Press, 1974.
    [2]
    DE JAGER E M, JIANG F. The Theory of Singular Perturbation[M]. Amsterdam: North-Holland Publishing Co, 1996.
    [3]
    NAYFEH A H. Introduction for Perturbation Techniques[M]. New York: John Wiley & Sons, 1981.
    [4]
    CHANG K W, HOWES F A. Nonlinear Singular Perturbation Phenomena: Theory and Application[M]. New York: Springer Verlag, 1984.
    [5]
    BOH A.The shock location for a class of sensitive boundary value problems[J]. J of Mathematical Analysis and Applications, 1999, 235(1): 295-314.
    [6]
    MO Jiaqi. The shock solutions for a class of singularly perturbed time delay boundary value problems[J]. Journal of Anhui Normal University (Natural Science), 2013, 36(4): 314-318.
    [7]
    XU Yonghong, SHI Lanfang, MO Jiaqi. Boundary perturbed problem for reaction diffusion time delay equation with two parameters[J]. Journal of Wuhan University (Natural Sciences), 2015, 20(2): 93-96.
    [8]
    MO Jiaqi, WANG Weigang, CHEN Xianfeng, et al. The shock wave solutions for singularly perturbed time delay nonlinear boundary value problems with two parameters[J]. Mathematica Applicata, 2014, 27(3): 470-475.
    [9]
    YAO Jingsun, MO Jiaqi. Singularly perturbed solution to semilinear higher order reaction diffusion equations with two parameters[J]. Annals of Differential Equations, 2009,25(1): 91-96.
    [10]
    FENG Yihu, MO Jiaqi. The shock asympototic solution for nonlinear elliptic equation with two parameters[J]. Mathematica Applicata, 2015, 28(3): 579-585.
    [11]
    朱红宝.一类非线性奇摄动时滞边值问题的激波解[J].中国科学技术大学学报,2018,48(5):357-360.
    ZHU Hongbao.The shock solution to a class of singularly perturbed time delay nonlinear boundary value problem[J].Journal of University of Science and Technology of China,2018,48(5):357-360.
    [12]
    TANG Rongrong. Solution with shock-boundary layer and shock-interior layer to a class of nonlinear problems[J]. Annals of Differential Equations, 2012, 28(1): 87-92.)
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