ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION

Qian Yang; Mingkang Ni

doi:10.11948/20210147

2022 Volume 12 Issue 1

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Qian Yang, Mingkang Ni. ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 256-269. doi: 10.11948/20210147

Citation:

Qian Yang, Mingkang Ni. ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 256-269. doi: 10.11948/20210147

ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION

Qian Yang¹,
Mingkang Ni^1,2, ,

1.
School of Mathematical Sciences, East China Normal University, No.500 Dongchuan Rd, 200241 Shanghai, China
2.
Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, No. 500 Dongchuan Rd, 200241 Shanghai, China

Corresponding author: Email: xiaovikdo@163.com(M. Ni)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 11871217) and the Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

Abstract

Abstract

We investigate a singularly perturbed boundary value problem for a piecewise-smooth second-order quasilinear differential equation in the case when the discontinuous curve which separates the domain is monotone. Applying the boundary layer function method, the asymptotic expansion of a solution with internal layer appearing in the neighborhoods of some point on the monotone curve and the point itself is constructed. For sufficiently small parameter values, using the matching method, the existence of a smooth solution with an internal transition layer in the neighborhood of a point of the monotone curve is proved. A simple example is given to show the effectiveness of our method.
- Quasilinear differential equation /
- internal layer /
- asymptotic method /
- piecewise-smooth dynamical system
MSC: 35B25, 35B40, 35B65, 35G30

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References

[1]	V. F. Butuzov, N. N. Nefedov and K. R. Schneider, Singularly perturbed elliptic problems in case of exchange of stability, Journal of Differential Equations, 2001, 169, 373-395. doi: 10.1006/jdeq.2000.3904 CrossRef Google Scholar
[2]	V. F. Butuzov, A. B. Vasil'eva and N. N. Nefedov, Asymptotic theory of contrasting structures, Automation and Remote Control, 1997, 58(7), 1068-1091. Google Scholar
[3]	C. A. Buzzi, P. R. da Silva and M. A. Teixeira, Slow-fast systems on algebraic varieties bordering piecewise-smooth dynamical systems, Bulletin Des Sciences Mathematiques, 2012, 136, 444-462. doi: 10.1016/j.bulsci.2011.06.001 CrossRef Google Scholar
[4]	H. Chen, Social status human capital formation and super-neutrality in a two sector monetary economy, Economic Modeling, 2011, 28, 785-794. doi: 10.1016/j.econmod.2010.10.010 CrossRef Google Scholar
[5]	Z. Du, J. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, Journal of Functional Analysis, 2018, 275, 988-1007. doi: 10.1016/j.jfa.2018.05.005 CrossRef Google Scholar
[6]	A. F. Filippov, Differential equations with discontinuous righthand sides, Springer, 1988. Google Scholar
[7]	G. Fusco and N. Guglielmi, A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type, Journal of Differential Equations, 2011, 250, 3230-3279. doi: 10.1016/j.jde.2010.12.013 CrossRef Google Scholar
[8]	Z. Guo and L. Huang, Global exponential convergence and global convergence in finite time of non-autonomous discontinuous neural networks, Nonlinear Dynamics, 2009, 58, 349-359. doi: 10.1007/s11071-009-9483-2 CrossRef Google Scholar
[9]	Z. Guo and L. Huang, LMI conditions for global robust stability of delayed neural networks with discontinuous neuron activations, Applied Mathematics and Computation, 2009, 215(3), 889-900. doi: 10.1016/j.amc.2009.06.013 CrossRef Google Scholar
[10]	Z. Guo, L. Huang, and X. Zou, Impact of discontinuous treatments on disease dynamics in an SIR epidemic model, Mathematical Biosciences and Engineering, 2012, 9(1), 97-110. doi: 10.3934/mbe.2012.9.97 CrossRef Google Scholar
[11]	J. W. Hargrove, J. H. Humphrey, A. Mahomva, et al, Declining HIV prevalence and incidence in perinatal women in Harare, Zimbabwe Epidemics, 2011, 3, 88-94. doi: 10.1016/j.epidem.2011.02.004 CrossRef Google Scholar
[12]	E. M. D. Jager and F. Jiang, The Theory of Singular Perturbations, Elsevier, North Holland, 1996. Google Scholar
[13]	F. Jiang and M. Han, Qualitative analysis of crossing limit cycles in discontinuous Liénard-type differential systems, Journal of Nonlinear Modeling and Analysis, 2019, 1(4), 527-543. Google Scholar
[14]	N. Levashova, A. Melnikova, A. Semina and A. Sidorova, Autowave mechanisms of structure formation in urban ecosystems as the process of self-organization in active media, Communication on Applied Mathematics and Computation, 2017, 31(1), 32-42. Google Scholar
[15]	N. T. Levashova, N. N. Nefedov and A. O. Orlov, Time-independent reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 2017, 57(5), 854-866. doi: 10.1134/S0965542517050062 CrossRef Google Scholar
[16]	N. T. Levashova, N. N. Nefedov and A. O. Orlov, Asymptotic stability of a stationary solution of a multidimensional reaction-diffusion equation with a discontinuous source, Computational Mathematics and Mathematical Physics, 2019, 59(4), 573-582. doi: 10.1134/S0965542519040109 CrossRef Google Scholar
[17]	X. Lin, J. Liu and C. Wang, The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems, Boundary Value Problems, 2020, 14, 1-17. Google Scholar
[18]	X. Lin, J. Liu and C. Wang, The existence and asymptotic estimates of solutions for a third-order nonlinear singularly perturbed boundary value problem, Qualitative Theory of Dynamical Systems, 2019, 18, 687-710. doi: 10.1007/s12346-018-0307-y CrossRef Google Scholar
[19]	D. V. Lukyanenko, M. A. Shishlenin and V. T. Volkov, Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time data, Communications in Nonlinear Science and Numerical Simulation, 2018, 54, 233-247. doi: 10.1016/j.cnsns.2017.06.002 CrossRef Google Scholar
[20]	N. Nefedov, The existence and asymptotic stability of periodic solutions with an interior layer of Burgers type equations with modular advection, Math. Model. Natl. Phenom., 2019, 4(4), 1-14. Google Scholar
[21]	N. N. Nefedov and M. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 2015, 55(12), 2001-2007. doi: 10.1134/S096554251512012X CrossRef Google Scholar
[22]	M. Ni, Y. Pang and N. T. Levashova, Internal layer for a system of singularly perturbed equations with discontinuous right-hand side, Differentsial'nye Uravneniya, 2018, 54(12), 1626-1637. Google Scholar
[23]	M. Ni, Y. Pang, N. T. Levashova and O. A. Nikolaeva, Internal layers for a singularly perturbed second-order quasilinear differential equation with discontinuous right-hand Side, Differential Equations, 2017, 53(12), 1616-1626. doi: 10.1134/S0012266117120096 CrossRef Google Scholar
[24]	O. E. Omel'chenko, L. Recke and V. F. Butuzov, Time-periodic boundary layer solutions to singularly perturbed parabolic problems, Journal of Differential Equations, 2017, 262(9), 4823-4862. doi: 10.1016/j.jde.2016.12.020 CrossRef Google Scholar
[25]	A. Orlov, N. Levashova and T. Burbaev, The use of asymptotic methods for modeling of the carriers wave functions in the Si/SiGe heterostructures with quantum-confined layers, Journal of Physics: Conference Series, 2015, 586(1), Article ID 012003. Google Scholar
[26]	A. B. Vasil'eva, Step-like contrasting structures for a singularly perturbed quasilinear second-order differential equation, Computational Mathematics and Mathematical Physics, 1995, 35(4), 411-419. Google Scholar
[27]	A. B. Vasil'eva and V. F. Butuzov, Asymptotic Methods in Singular Perturbation Theory, Moscow: Vysshaya Shkola, 1990. Google Scholar
[28]	A. B. Vasil'eva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbed Problem, SIAM Philadelphia, 1995. Google Scholar
[29]	A. B. Vasil'eva, V. F. Butuzov and N. N. Nefedov, Contrasting structures in singularly perturbed problems, Fundamentalnaya i Prikladnaya Matematika, 1998, 4(3), 799-851. Google Scholar
[30]	A. B. Vasil'eva, V. F. Butuzov and N. N. Nefedov, Singularly perturbed problems with boundary and internal layers, Proceedings of the Steklov Institute of Mathematics, 2010, 268, 258-273. doi: 10.1134/S0081543810010189 CrossRef Google Scholar
[31]	V. T. Volkov, D. V. Luk'yanenko and N. N. Nefedov, Analytical-numerical approach to describing timeperiodic motion of fronts in singularly perturbed reaction-advection-diffusion models, Computational Mathematics and Mathematical Physics, 2019, 59(1), 46-58. doi: 10.1134/S0965542519010159 CrossRef Google Scholar
[32]	V. T. Volkov and N. N. Nefedov, Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reaction-diffusion equations, Computational Mathematics and Mathematical Physics, 2006, 46(4), 585-593. doi: 10.1134/S0965542506040075 CrossRef Google Scholar
[33]	C. Wang and X. Zhang, Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type Ⅲ, Journal of Differential Equations, 2019, 267, 3397-3441. doi: 10.1016/j.jde.2019.04.008 CrossRef Google Scholar
[34]	Z. Zhou and J. Shen, Delayed phenomenon of loss of stability of solutions in a second-order quasi-linear singularly perturbed boundary value problem with a turning point, Boundary Value Problems, 2011, 2011(1), 1-13. doi: 10.1186/1687-2770-2011-1 CrossRef Google Scholar