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

FENG Yi-hu; LIU Shu-de; MO Jia-qi

doi:10.21656/1000-0887.370177

Volume 38 Issue 5

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2017 > 38(5): 561-569

FENG Yi-hu, LIU Shu-de, MO Jia-qi. Generalized Solution to a Class of Singularly Perturbed Problem of Nonlinear Reaction Diffusion Equation With Two Parameters[J]. Applied Mathematics and Mechanics, 2017, 38(5): 561-569. doi: 10.21656/1000-0887.370177

Citation:

PDF( 341 KB)

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

doi: 10.21656/1000-0887.370177

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)

Received Date: 2016-06-06
Rev Recd Date: 2016-10-01
Publish Date: 2017-05-15

Abstract

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

FullText(HTML)

References(19)

References

[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))