THE INTERIOR LAYER FOR A NONLINEAR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION

doi:10.1016/S0252-9602(12)60049-6

数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (2): 695-709.doi: 10.1016/S0252-9602(12)60049-6

THE INTERIOR LAYER FOR A NONLINEAR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION

王爱峰^1,2,3|倪明康^1,3

1. Department of Mathematics, East China Normal University, Shanghai 200062, China
2. School of Mathematical Science, Huaiyin Normal University, Huaian 223001, China
3. Division of Computational Science, E-Institute of Shanghai Universities at SJTU, Shanghai 200030

收稿日期:2010-09-23 修回日期:2011-03-12 出版日期:2012-03-20 发布日期:2012-03-20
基金资助:
Supported by the National Natural Science Funds (11071075), the Natural Science Foundation of Shanghai (10ZR1409200), the National Laboratory of Biomacro-molecules, Institute of Biophysics, Chinese Academy of Sciences, and the E-Institutes of Shanghai Municipal
Education Commissions (E03004).

THE INTERIOR LAYER FOR A NONLINEAR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION

WANG Ai-Feng^1,2,3, NI Ming-Kang^1,3

1. Department of Mathematics, East China Normal University, Shanghai 200062, China
2. School of Mathematical Science, Huaiyin Normal University, Huaian 223001, China
3. Division of Computational Science, E-Institute of Shanghai Universities at SJTU, Shanghai 200030

Received:2010-09-23 Revised:2011-03-12 Online:2012-03-20 Published:2012-03-20
Supported by:
Supported by the National Natural Science Funds (11071075), the Natural Science Foundation of Shanghai (10ZR1409200), the National Laboratory of Biomacro-molecules, Institute of Biophysics, Chinese Academy of Sciences, and the E-Institutes of Shanghai Municipal
Education Commissions (E03004).

摘要/Abstract

摘要：

In this article, the interior layer for a second order nonlinear singularly per-turbed differential-difference equation is considered. Using the methods of boundary func-tion and fractional steps, we construct the formula of asymptotic expansion and point out that the boundary layer at t = 0 has a great influence upon the interior layer at t = σ. At the same time, on the basis of differential inequality techniques, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of our result. The result of this article is new and it complements the previously known ones.

关键词: Differential-difference equation, interior layer, asymptotic expansion, bound-ary function

Abstract:

Key words: Differential-difference equation, interior layer, asymptotic expansion, bound-ary function

中图分类号:

34E20

王爱峰, 倪明康. THE INTERIOR LAYER FOR A NONLINEAR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION[J]. 数学物理学报(英文版), 2012, 32(2): 695-709.

WANG Ai-Feng, NI Ming-Kang. THE INTERIOR LAYER FOR A NONLINEAR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION[J]. Acta mathematica scientia,Series B, 2012, 32(2): 695-709.

参考文献

[1] Tzou D Y. Micro-to-macroscale Heat Transfer, Taylor and Francis. Washington, DC. 1997

[2] Liu B W. Almost periodic solutions for hopfield neural networks with continuously distributed delays. Math Comput Simulation, 2007, 73: 327–335

[3] Ramos J I. Exponetial methods for singularly-perturbed ordinary differential-difference equations. Appl Math Comput, 2006, 182(2): 1528–1541

[4] Patidar K C, Sharma K K. Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay or advance. Internat J Numer Methods Engry, 2006, 66(2): 272–296

[5] Lange C G, Miura R M. Singular perturbation analysis of boundary-value problems for differential differ-ence equations. SIAM Journal on Applied Mathematics, 1982, 42: 502–531

[6] Lange C G, Miura R M. Singular perturbation analysis of boundary-value problems for differential differ-ence equations. II. Rapid oscillations and resonances, SIAM Journal on Applied Mathematics, 1985, 45: 687–707

[7] Lange C G, Miura R M. Singular perturbation analysis of boundary-value problems for differential differ-ence equations. III. Turning point problems. SIAM Journal on Applied Mathematics, 1985, 45: 708–734

[8] Lange C G, Miura R M. Singular perturbation analysis of boundary-value problems for differential differ-ence equations. IV. Small shift with rapid oscillations. SIAM Journal on Applied Mathematics, 1994, 54: 273–283

[9] Lange C G, Miura R M. Singular perturbation analysis of boundary-value problems for differential dif-ference equations. V. Small shift with layer behavior. SIAM Journal on Applied Mathematics, 1994, 54: 249–272

[10] Kadalbajoo M K, Sharma K K, Devendra Kumar. Numerical analysis of singularly perturbed delay differential equations with layer behavior. Appl Math Comput, 2004, 157: 11–28

[11] Kadalbajoo M K, Devendra Kumar. Fitted mesh B-spline collocation method for singularly perturbed differential-difference equations with small delay. Appl Math Comput, 2008, 204: 90–98

[12] Miao S M, Zhou Q D. Bundary Value Problems for Singular Perturbed Differential-difference Equations. Jilin Univ Natur Sci Ed, 1987, 3: 1–7

[13] Tian H J. Asymptotic expansion for the solution of singularly perturbed delay differential equations. Math Anal Appl, 2003, 281(2): 678–696

[14] Kadalbajoo M K, Sharma K K. Parameter uniform numercial method for a boundary-value problem for singular perturted nonlinear delay differential equation of neutral type. Int J Comput Math, 2004, 81(7): 845–862

[15] Lange C G. Miura R M. Singular perturbation analysis of boundary-value problems for differential differ-ence equations. IV. A nonlinear example with layer behavior. Studies Appl Math, 1991, 84: 231–273

[16] Ni M K, Lin W Z. The asymptotic solutions of delay singularly perturbed differential difference equations. Acta Mathematica Scientia, 2010, 30A(6): 1413–1423

[17] Vasileva A B, Buluzov V F. Ni M K, Lin W Z. Asymptotic expansions of singularly perturbed differential equations. Beijing: Higher Education Press, 2008 (in China)

THE INTERIOR LAYER FOR A NONLINEAR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION

THE INTERIOR LAYER FOR A NONLINEAR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATION

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