ON THE ASYMPTOTIC SOLUTION TO A TYPE OF PIECEWISE-CONTINUOUS SECOND-ORDER DIRICHLET PROBLEMS OF TIKHONOV SYSTEM

Xutian Qi; Mingkang Ni

doi:10.11948/2019.105

2019 Volume 9 Issue 1

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Xutian Qi, Mingkang Ni. ON THE ASYMPTOTIC SOLUTION TO A TYPE OF PIECEWISE-CONTINUOUS SECOND-ORDER DIRICHLET PROBLEMS OF TIKHONOV SYSTEM[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 105-117. doi: 10.11948/2019.105

Citation:

Xutian Qi, Mingkang Ni. ON THE ASYMPTOTIC SOLUTION TO A TYPE OF PIECEWISE-CONTINUOUS SECOND-ORDER DIRICHLET PROBLEMS OF TIKHONOV SYSTEM[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 105-117. doi: 10.11948/2019.105

ON THE ASYMPTOTIC SOLUTION TO A TYPE OF PIECEWISE-CONTINUOUS SECOND-ORDER DIRICHLET PROBLEMS OF TIKHONOV SYSTEM

Xutian Qi¹,
Mingkang Ni^1, ,

1.
School of Mathematical Sciences, East China Normal University, No.500 Dongchuan Rd, 200241 Shanghai, China

Corresponding author: Email address:xiaovikdo@163.com(M. Ni)
Fund Project: The authors were supported by National Natural Science Foundation of China (11471118, 11871217) and were supported in part by Science and Technology Commission of Shanghai Municipality

Abstract

Abstract

A Type of second-order Dirichlet problems of Tikhonov system with piecewise-continuous right hand side is studied. By using the multiscale theory and the theory of contrast structures, a first-order continuous, uniform and effective asymptotic solution of the problem is constructed. Existence of the solution is proved and the remainder is estimated. An illustrative example for explaining this method is also given.
- Tikhonov System /
- theory of contrast structures /
- piecewise continuous
MSC: 93C70

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References

[1]	V. F. Butuzov, A. B. Vasil'eva and N. N. Nefedov, Asymptotic theory of contrasting structures, Automation & Remote Control, 1997, 58(7), 1068-1091. Google Scholar
[2]	C. A. Buzzi, P. R. da Silva and M. A. Teixeira, A Singular approach to discontinuous vector fields on the plane, Journal of Differential Equations, 2006, 231(2), 633-655. doi: 10.1016/j.jde.2006.08.017 CrossRef Google Scholar
[3]	Y. Cheng and Q. Zhang, Local analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem, 2017, 72(2), 792-819. Google Scholar
[4]	W. Lin, Z. Wang and J. Zhang, Ordinary Differential Equations, Science Press, Beijing, 2003. Google Scholar
[5]	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
[6]	N. N. Nefedov and M. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics & Mathematical Physics, 2015, 55(12), 2001-2007. Google Scholar
[7]	M. Ni and W. Ni, The Asymptotic theory of singular perturbation problems, Higher Education Press, Beijing, 2009. Google Scholar
[8]	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
[9]	T. Prabha, M. Chandru and V. Shanthi, Hybird difference scheme for singularly perturbed reaction-convection-diffusion problem with boundary and interior layers, Applied Mathematics and Computation, 2017, 314, 237-256. doi: 10.1016/j.amc.2017.06.029 CrossRef Google Scholar
[10]	A. B. Vasil'eva and V. F. Butuzov, Asymptotic expansions of solutions of singularly perturbed equations(in Russian), Nauka, Moscow, 1973. Google Scholar
[11]	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(1), 258-273. doi: 10.1134/S0081543810010189 CrossRef Google Scholar
[12]	A. B. Vasil'eva, V. F. Butuzov and N. N. Nefedov, Contrasting structures in singularly perturbed problems, Fundam. Prikl. Mat, 1998, 4(3), 799-851. Google Scholar
[13]	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(1), 258-273. doi: 10.1134/S0081543810010189 CrossRef Google Scholar
[14]	A. B. Vasil'eva and V. F. Butuzov, Asymptotic methods of singular perturbation theory(in Russian), Vysshaya Shkola, Moscow, 1990. Google Scholar
[15]	A. B. Vasil'eva and V. F. Butuzov, Asymptotic methods in singularly perturbed problems(in Russian), Yaroslavl, 2014. Google Scholar