Abstract
This article is reporting study of the solution for a singularly perturbed nonlinear system of differential equations. In this paper, solutions of a singularly perturbed nonlinear system are considered in a particularly critical case. The matrix of a linear system has complex conjugate eigenvalues. The eigenvalues of the matrix of the system under consideration do not have zeros on the boundary of the region under consideration and outside this region, and the imaginary parts of the eigenvalues of the matrix are positive with the exception of boundary points in the considered domain. A uniform approximation was constructed for the solution of the initial Cauchy problem in a particularly critical case with any degree of accuracy. Results were also obtained for a specific task.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Kaur, M., Arora, G.: A review on singular perturbed delay differential equations. Int. J. Curr. Adv. Res. 6(3), 2341–2346 (2017). https://doi.org/10.24327/ijcar.2017.2346.0005
Proctor, T.G.: Perturbed nonlinear differential equations. J. Math. Anal. Appl. 45(3), 777–791 (1974). https://doi.org/10.1016/0022-247X(74)90067-5
Yan, X.: Singularly Perturbed Differential/Algebraic Equations. I: Asymptotic Expansion of Outer Solutions. J. Mathem. Anal. Appl. 207(2), 326–344 (1997). https://doi.org/10.1006/jmaa.1997.5256
Voulov, H.D., Bainov, D.D.: Asymptotic stability of the solutions of a linear singularly perturbed system with unbounded delay. J. Math. Anal. Appl. 155(2), 293–311 (1991). https://doi.org/10.1016/0022-247X(91)90001-G
Bansal, K., Sharma, K.K.: Parameter-robust numerical scheme for time-dependent singularly perturbed reaction-diffusion problem with large delay. Numer. Func. Anal. Optim. 39(2), 127–154 (2018). https://doi.org/10.1080/00207160410001708823
Kalaba, R., Spingarn, K., Tesfatsion: Variational equations for the eigenvalues and eigenvectors of nonsymmetric matrices. L. J. Optim. Theory Appl. 33: 1 (1981). https://doi.org/10.1007/BF00935172
Liu, K.M.: Numerical solution of differential eigenvalue problems with variable coefficients with the Tau-Collocation method. Math. Comput. Model. 11, 672–675 (1988). https://doi.org/10.1016/0895-7177(88)90577-8
Allahviranloo, T., Gouyandeh, Z., Armand, A.: Numerical solutions for fractional differential equations by Tau-Collocation method. Appl. Math. Comput. 271, 979–990 (2015). https://doi.org/10.1016/j.amc.2015.09.062
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Karimov, S.K., Anarbaeva, G.M. (2021). Uniform Approximations to Solutions of Singularly Perturbed Systems of Differential Equations with the Eigenvalues Which Have No Zero in the Region Under Consideration. In: Popkova, E.G., Ostrovskaya, V.N., Bogoviz, A.V. (eds) Socio-economic Systems: Paradigms for the Future. Studies in Systems, Decision and Control, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-56433-9_72
Download citation
DOI: https://doi.org/10.1007/978-3-030-56433-9_72
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-56432-2
Online ISBN: 978-3-030-56433-9
eBook Packages: EngineeringEngineering (R0)