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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (4): 419-427.doi: 10.3969/j.issn.1005-3085.2016.04.008

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具有两参数的非线性非局部扰动发展方程奇摄动问题(英)

冯依虎1,  吴钦宽2,  许永红3,  莫嘉琪4   

  1. 1- 亳州学院电子与信息工程系，安徽 亳州 236800
    2- 南京工程学院数理部，江苏  南京 211167
    3- 蚌埠学院数理系，安徽 蚌埠 233030
    4- 安徽师范大学数学系，安徽 芜湖 241003
  • 收稿日期:2014-12-18 接受日期:2015-12-07 出版日期:2016-08-15 发布日期:2016-10-15
  • 基金资助:
    国家自然科学基金 (11202106)；安徽省教育厅自然科学基金 (KJ2015A347; KJ2014A151; KJ2013B153)；安徽省高等学校优秀青年人才项目 (gxyqZD2016520).

The Singularly Perturbed Problems for Nonlinear Nonlocal Disturbed Evolution Equations with Two Parameters

FENG Yi-hu1,  WU Qin-kuan2,  XU Yong-hong3,  MO Jia-qi4   

  1. 1- Department of Electronics and Information Engineering, Bozhou College, Bozhou, Anhui 236800
    2- Department of Mathematics & Physics, Nanjing Institute of Technology, Nanjing, Jiangsu 211167
    3- Department of Mathematics & Physics, Bengbu College, Bengbu, Anhui 233030
    4- Department of Mathematics, Anhui Normal University, Wuhu, Anhui 241003
  • Received:2014-12-18 Accepted:2015-12-07 Online:2016-08-15 Published:2016-10-15
  • Supported by:
    The National Natural Science Foundation of China (11202106); the Natural Science Foundation of the Education Department of Anhui Province (KJ2015A347; KJ2014A151; KJ2013B153); the Excellent Youth Talented Project of the Colleges and Universities of Anhui Province (gxyqZD2016520).

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摘要： 本文研究了一类具有非线性非局部扰动发展方程的奇摄动问题．利用奇摄动方法，对该问题解的结构在两个小参数相互关联的情形和适当的条件下展开了全面研究．首先，构造了问题的外部解；然后在区域的边界邻域构造局部坐标系，再在该邻域中引入多尺度变量，得到问题解的边界层校正项，另外，通过引入伸长变量构造了初始层校正项；最后，利用不动点定理，证明了问题的解的一致有效的渐近展开式．用上述方法得到的各次近似解，具有便于求解、精度高等特点．

关键词: 非局部问题, 奇摄动, 非线性双曲型方程

Abstract:

The nonlinear nonlocal singularly perturbed problems for the disturbed evolution equations are studied. Using the singular perturbation method, the structure of solution to the problem is discussed in the cases of two small parameters. Under the suitable conditions, firstly, the outer solution to the boundary value problem is given. Secondly, the variables of multiple scales are introduced to obtain the boundary layer corrective terms for the solution. Then the stretched variable is applied to the boundary neighborhood to get the initial layer correction term. Finally, using the fix point theorem, the uniformly valid asymptotic expansion of the solution to the problem is proved. The proposed method possesses the advantages of convenient use and high accuracy.

Key words: nonlocal problem, singular perturbation, nonlinear hyperbolic equation

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