Journal of East China Normal University(Natural Science) >

2017 >Issue 3: 39 - 47

DOI: https://doi.org/10.3969/j.issn.1000-5641.2017.03.004

Mathematics

A class of singularly perturbed hyperbolic nonlinear integral-differential system

  • FENG Yi-hu ,
  • MO Jia-qi
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  • 1. Department of Electronics and Information Engineering, Bozhou College, Bozhou Anhui 236800, China;
    2. Department of Mathematics, Anhui Normal University, Wuhu Anhui 241003, China

Received date: 2016-03-17

  Online published: 2017-05-18

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Abstract

A class of singularly perturbed system for the hyperbolic nonlinear integral-differential system is considered. Firstly, the outer solution to system is obtained by employing the Fredholm type integral equation. Then the boundary layer corrective term is constructed using the variables of multiple scales method. And the initial layer corrective term is found via the stretched variable method. Finally, from the fixed point theory, the uniformly valid behavior for the composed asymptotic expansion of singular perturbation solution is proved.

Key words： integral-differential equation; singular perturbation; hyperbolic type equation

Cite this article

FENG Yi-hu , MO Jia-qi . A class of singularly perturbed hyperbolic nonlinear integral-differential system[J]. Journal of East China Normal University(Natural Science), 2017 , (3) : 39 -47 . DOI: 10.3969/j.issn.1000-5641.2017.03.004

References

[1] DE JAGER E M, JIANG F R. The Theory of Singular Perturbation[M]. Amsterdam: North-Holland Publishing Co, 1996.
[2] BARBU L, MOROSANU G. Singularly Perturbed Boundary-Value Problems[M]. Basel: Birkhauser, 2007.
[3] CHANG KW, HOWES F A. Nonlinear Singular Perturbation Phenomena: Theory and Applications [M]. Applied Mathematical Science, 56, New York: Springer-Verlag, 1984.
[4] SAMUSENKO P F. Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations [J]. J Math Sci, 2013, 189 (5): 834-847.
[5] MARTINEZ S, WOLANSKI N. A singular perturbation problem for a quasi-linear operator satisfying the natural condition of Lieberman [J]. SIAM J Math Anal, 2009, 41(1): 318-359.
[6] KELLOGG R B, KOPTEVA N A. Singularly perturbed semilinear reaction-diffusion problem in a polygonal domain[J]. J Differ Equations, 2010, 248(1): 184-208.
[7] TIAN C R, ZHU P. Existence and asymptotic behavior of solutions for quasilinear parabolic systems [J]. Acta Appl Math, 2012, 121(1): 157-173.
[8] SKRYNNIKOV Y. Solving initial value problem by matching asymptotic expansions[J]. SIAM J Appl Math, 2012, 72(1): 405-416.
[9] KELLY W G. A singular perturbation problem of Carrier and Pearson[J]. J Math Anal Appl, 2001, 255: 678-697.
[10] MIZOGUCHI N, YANAGIDA E. Life span of solutions for a semilinear parabolic problem with small diffusion[J]. J Math Anal Appl, 2001, 261: 350-368.
[11] MO J Q. Singular perturbation for a boundary value problem of fourth order nonlinear differential equation [J]. Chin Ann Math B, 1987(1): 80-88.
[12] MO J Q. Singular perturbation for a class of nonlinear reaction diffusion systems[J]. Science in China, 1989, 32: 1306-1315.
[13] MO J Q. A singularly perturbed nonlinear boundary value problem [J]. J Math Ana1 Appl, 1993, 178: 289-293.
[14] MO J Q. Homotopic mapping solving method for gain fluency of a laser pulse amplifier [J]. Science in China G, 2009, 52(7): 1007-1010.
[15] FENG Y H, LIU S D. Spike layer solutions of some quadratic singular perturbation problems with high-order turning points [J]. Math Appl, 2014, 27(1): 50-55.
[16] 冯依虎, 石兰芳, 汪维刚, 等. 一类广义非线性强阻尼扰动发展方程的行波解[J]. 应用数学和力学, 2015, 36(3): 315-324.
[17] 冯依虎, 石兰芳, 汪维刚, 等. 一类大气尘埃等离子体扩散模型研究[J]. 应用数学和力学, 2015, 36(6): 639-650.
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