一类两参数非线性反应扩散方程奇摄动问题的广义解

冯依虎; 刘树德; 莫嘉琪

doi:10.21656/1000-0887.370177

一类两参数非线性反应扩散方程奇摄动问题的广义解

doi: 10.21656/1000-0887.370177

¹亳州学院电子与信息工程系, 安徽亳州 236800;²安徽师范大学数学计算机科学学院, 安徽芜湖 241003

基金项目: 国家自然科学基金(11202106);安徽省教育厅自然科学重点基金(KJ2015A347;KJ2017A702);安徽省高校优秀青年人才支持计划重点项目(gxyqZD2016520)

详细信息

作者简介:
冯依虎（1982—），男，副教授，硕士(E-mail: fengyihubzsz@163.com)；莫嘉琪(1937—)，男，教授(通讯作者. E-mail: mojiaqi@mail.ahnu.edu.cn).

中图分类号: O175.29
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出版历程
- 收稿日期: 2016-06-06
- 修回日期: 2016-10-01
- 刊出日期: 2017-05-15

Generalized Solution to a Class of Singularly Perturbed Problem of Nonlinear Reaction Diffusion Equation With Two Parameters

¹Department of Electronics and Information Engineering, Bozhou University, Bozhou, Anhui 236800, P.R.China;²School of Mathematics & Computer Science, Anhui Normal University, Wuhu, Anhui 241003, P.R.China

Funds: The National Natural Science Foundation of China(11202106)

摘要

摘要: 利用奇异摄动方法讨论了一类两参数广义奇摄动反应扩散方程问题.首先，在适当的条件下，对两个小参数进行幂级数展开，构造了问题的形式外部解.其次，在区域边界邻近，建立局部坐标系，利用多重尺度变量方法分别构造了问题解的第一、第二边界层校正项.最后，利用合成展开理论，得到了问题广义解的渐近表示式，并用泛函分析不动点原理，估计了渐近展开式的精度.该文得到问题的广义解在重叠区域内具有两个不同厚度的校正函数.它们分别对边界条件起着校正的作用，扩展了问题研究范围，同时还提供了构造这类在重叠区域上不同厚度的校正项的方法，因此具有广泛的研究前景.
- 反应扩散 /
- 奇异摄动 /
- 广义解
Abstract: A class of generalized singularly perturbed problems of reaction diffusion equations with two parameters were considered with the singular perturbation method. Firstly, under suitable conditions, the outer solution to the problem was found. Next, the power series of the two small parameters were developed, and the first and second boundary layer corrective terms for the solution to the problem were constructed with the multiscale variable method, respectively. Finally, based on the composite expansion method, the asymptotic expression of the generalized solution to the problem was obtained, and according to the fixed point theory for functional analysis, the precision of the asymptotic expansion was estimated. Two corrective functions with different thicknesses were obtained for the generalized solution in the overlapping area, and they take effects on the boundary conditions respectively and expand the range of study; moreover, the work provides a costruction method for this kind of corrective terms with different thicknesses in the overlapping area, thus has a wide study foreground.
- reaction diffusion /
- singular perturbation /
- generalized solution

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参考文献(19)

[1]	de Jager E M, Furu J. The Theory of Singular Perturbation [M]. Amsterdam: North-Holland Publishing Co, 1996.
[2]	Barbu L, Morosanu G. Singularly Perturbed Boundary-Value Problems [M]. Basel: Birkhauserm Verlag AG, 2007.
[3]	Martínez S, Wolanski N. A singular perturbation problem for a quasi-linear operator satisfying the natural growth condition of Lieberman[J]. SIAM Journal on Mathematical Analysis,2009,41(1): 318-359.
[4]	Kellogg R B, Kopteva N A. Singularly perturbed semilinear reaction-diffusion problem in a polygonal domain[J]. Journal of Differential Equations, 2010,248(1): 184-208.
[5]	Tian C, Zhu P. Existence and asymptotic behavior of solutions for quasilinear parabolic systems[J]. Acta Applicandae Mathematicae, 2012,121(1): 157-173.
[6]	Skrynnikov Y. Solving initial value problem by matching asymptotic expansions[J]. SIAM Journal on Applied Mathematics, 2012,72(1): 405-416.
[7]	Samusenko P. Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations[J]. Journal of Mathematical Sciences, 2013,189(5): 834-847.
[8]	MO Jia-qi, HAN Xiang-lin, CHEN Song-lin. The singularly perturbed nonlocal reaction diffusion system[J]. Acta Mathematica Scientia, 2002,22B(4): 549-556.
[9]	MO Jia-qi, HAN Xiang-lin. A class of singularly perturbed generalized solution for the reaction diffusion problems[J]. Journal of Systems Science and Mathematical Sciences, 2002,22(4): 447-451.
[10]	MO Jia-qi, LIN Wan-tao. A class of nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation[J]. Acta Mathematicae Applicatae Sinica, 2006,22(1): 27-32.
[11]	MO Jia-qi. A class of singularly perturbed differential-difference reaction diffusion equation[J]. Advances in Mathematics, 2009,38(2): 227-231.
[12]	MO Jia-qi. Homotopic mapping solving method for gain fluency of a laser pulse amplifier[J]. Science in China(Ser G): Physics, Mechanics & Astronomy, 2009, 52(7): 1007-1070.
[13]	MO Jia-qi, LIN Wan-tao. Asymptotic solution of activator inhibitor systems for nonlinear reaction diffusion equations[J]. Journal of Systems Science and Complexity,2008,20(1): 119-128.
[14]	MO Jia-qi. A singularly perturbed reaction diffusion problem for the nonlinear boundary condition with two parameters[J]. Chinese Physics B,2010,19(1): 010203.
[15]	FENG Yi-hu, MO Jia-qi. The shock asymptotic solution for nonlinear elliptic equation with two parameters[J]. Mathematica Applicata, 2015,27(3): 579-585.
[16]	FENG Yi-hu, MO Jia-qi. Asymptotic solution for singularly perturbed fractional order differential equation[J]. Journal of Mathematics,2016,36(2): 239-245.
[17]	FENG Yi-hu, CHEN Xian-feng, MO Jia-qi. The generalized interior shock layer solution of a class of nonlinear singularly perturbed reaction diffusion problem[J]. Mathematica Applicata,2016,29(1): 161-165.
[18]	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.(FENG Yi-hu, SHI Lan-fang, WANG Wei-gang, et al. Traveling wave solution to a class of generalized nonlinear strong-damping disturbed evolution equations[J]. Applied Mathematics and Mechanics,2015,36(3): 315-324.(in Chinese))
[19]	冯依虎, 莫嘉琪. 一类非线性非局部扰动LGH方程的孤子行波解[J]. 应用数学和力学, 2016,37(4): 426-433.(FENG Yi-hu, MO Jia-qi. Soliton travelling wave solutions to a class of nonlinear nonlocal disturbed LGH equations[J]. Applied Mathematics and Mechanics, 2016,37(4): 426-433.(in Chinese))

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一类两参数非线性反应扩散方程奇摄动问题的广义解

doi: 10.21656/1000-0887.370177

作者简介:
冯依虎（1982—），男，副教授，硕士(E-mail: fengyihubzsz@163.com)；莫嘉琪(1937—)，男，教授(通讯作者. E-mail: mojiaqi@mail.ahnu.edu.cn).

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Generalized Solution to a Class of Singularly Perturbed Problem of Nonlinear Reaction Diffusion Equation With Two Parameters

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留言板

一类两参数非线性反应扩散方程奇摄动问题的广义解

doi: 10.21656/1000-0887.370177

作者简介: 冯依虎（1982—），男，副教授，硕士(E-mail: fengyihubzsz@163.com)；莫嘉琪(1937—)，男，教授(通讯作者. E-mail: mojiaqi@mail.ahnu.edu.cn).

计量

出版历程

Generalized Solution to a Class of Singularly Perturbed Problem of Nonlinear Reaction Diffusion Equation With Two Parameters

计量

出版历程

目录

作者简介:
冯依虎（1982—），男，副教授，硕士(E-mail: fengyihubzsz@163.com)；莫嘉琪(1937—)，男，教授(通讯作者. E-mail: mojiaqi@mail.ahnu.edu.cn).