A Class of Fractional Nonlinear Singularly Perturbed Problems With Time Delays

ZHU Hongbao

doi:10.21656/1000-0887.400195

Volume 40 Issue 12

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2019 > 40(12): 1356-1363

ZHU Hongbao. A Class of Fractional Nonlinear Singularly Perturbed Problems With Time Delays[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1356-1363. doi: 10.21656/1000-0887.400195

Citation:

ZHU Hongbao. A Class of Fractional Nonlinear Singularly Perturbed Problems With Time Delays[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1356-1363. doi: 10.21656/1000-0887.400195

Citation:

ZHU Hongbao. A Class of Fractional Nonlinear Singularly Perturbed Problems With Time Delays[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1356-1363. doi: 10.21656/1000-0887.400195

PDF( 330 KB)

A Class of Fractional Nonlinear Singularly Perturbed Problems With Time Delays

doi: 10.21656/1000-0887.400195

ZHU Hongbao

School of Mathematics & Physics, Anhui University of Technology, Maanshan, Anhui 243002, P.R.China

Received Date: 2019-06-20
Rev Recd Date: 2019-08-10
Publish Date: 2019-12-01

Abstract

Abstract

A class of fractional nonlinear singularly perturbed problems with time delays were considered. Firstly, the outer solution was constructed by means of the singular perturbation method. Then, a stretched variable was introduced to obtain 2 boundary layer correction items for the solution, and the asymptotic analytic expansion solution to the problem was also acquired. Finally, under suitable conditions, the theory of differential inequalities was applied to prove the uniformly valid asymptotic expansion of the solution to the original problem, and the conclusion with the future research directions was given.
- fractional differential equation,
- nonlinear,
- time delay,
- singularly perturbed

FullText(HTML)

References(17)

References

[1]	NAYFEH A H. Introduction to Perturbation Techniques [M]. New York: John Wiley & Sons Inc, 1981.
[2]	DE JAGER E M, FURU J F. The Theory of Singular Perturbation [M]. Amsterdam: North-Holland Publishing Co, 1996.
[3]	BOH A. The shock solution for a class of sensitive boundary value problems[J]. Journal of Mathematical Analysis and Applications,1999,235(1): 295-314.
[4]	倪明康, 林武忠. 边界层函数法在微分不等式中的应用[J]. 华东师范大学学报(自然科学版), 2007(3): 1-10.(NI Mingkang, LIN Wuzhong. Application of boundary layer function method in differential inequality[J]. Journal of East China Normal University(Natural Science),2007(3): 1-10.(in Chinese))
[5]	葛志新, 陈咸奖, 陈松林. 一类含有分数阶导数的二自由度耦合系统[J]. 应用数学和力学, 2017,38(11): 1300-1308.(GE Zhixin, CHEN Xianjiang, CHEN Songlin. A class of 2-DOF coupled systems with fractional-order derderivatives[J].Applied Mathematics and Mechanics,2017,38(11): 1300-1308.(in Chinese))
[6]	冯依虎, 陈怀军, 莫嘉琪. 一类非线性奇异摄动自治微分系统的渐近解[J]. 应用数学和力学, 2018,39(3): 355-363.(FENG Yihu, CHEN Huaijun, MO Jiaqi. Asymptotic solution to a class of nonlinear singular perturbation autonomous differential systems[J]. Applied Mathematics and Mechanics,2018,39(3): 355-363.(in Chinese))
[7]	莫嘉琪. 非线性分数阶微分方程的奇摄动[J]. 应用数学学报, 2006,29(6): 1085-1089.(MO Jiaqi. Singularly perturbed problems for nonlinear fractional differential equation[J]. Acta Mathematicae Applicatae Sinica,2006,29(6): 1085-1089.(in Chinese))
[8]	SHI J R, MO J Q. Asymptotic solution for a class of singularly perturbed initial value problem of fractional differential equation[J]. Acta Scientiarum Naturalium Universitatis Nankaiensis,2015,48(5): 60-64.
[9]	林学渊, 谢峰. 一类非线性分数阶微分方程的奇异摄动[J]. 东华大学学报(自然科学版), 2009,35(2): 238-240.(LIN Xueyuan, XIE Feng. Singular perturbation for a kind of nonlinear fractional differential equations[J]. Journal of Donghua University(Natural Science),2009,35(2): 238-240.(in Chinese))
[10]	莫嘉琪, 温朝晖. 一类非线性奇摄动分数阶微分方程的渐近解[J]. 系统科学与数学, 2010,30(12): 1689-1694.(MO Jiaqi, WEN Zhaohui. Asymptotic solution for a class of nonlinear singularly perturbed for fractional differential equation[J]. Journal of Systems Science and Mathematical Sciences,2010,30(12): 1689-1694.(in Chinese))
[11]	FENG Y H, MO J Q. Asymptopic solution for singularly perturbed fractional order differential equation[J]. Journal of Mathematic,2016,36(2): 239-245.
[12]	WANG W K, SHI L F, HAN X L, et al. Singular perturbation problem for reaction diffusion time delay equation with boundary perturbation[J]. Chinese Journal of Engineering Mathematics,2015,32(2): 291-297.
[13]	MO J Q, WANG W G, CHEN X G, et al. The shock wave solutions for singularly perturbed time delay nonlinear boundary value problems with two parameters[J]. Mathematica Applicata,2014,27(3): 470-475.
[14]	MO J Q. The shock solutions for a class of singularly perturbed time delay boundary value problems[J]. Journal of Anhui Normal University(Natural Science),2013,36(4): 314-318.
[15]	欧阳成. 具有小延迟的微分-差分方程渐近解[J]. 吉林大学学报(理学版), 2008,46(4): 628-932.(OUYANG Cheng. Asymptotic solution of initial value problems for differential-difference equation with small time delay[J]. Journal of Jilin University(Science Edition),2008,46(4): 628-632.(in Chinese))
[16]	DELBOSCO D, RODINO L. Existence and uniqueness for nonlinear fractional differential equation[J]. Journal of Mathematical Analysis and Applications,1996,204: 609-625.
[17]	莫嘉琪. 一类两参数半线性奇摄动问题解的渐近性态[J]. 应用数学学报, 2009,32(5): 903-908.(MO Jiaqi. The Asymptotic behavior of solution for a class of semilinear singular perturbed problem with two parameters[J]. Acta Mathematicae Applicatae Sinica,2009,32(5): 903-908.(in Chinese))