# zbMATH — the first resource for mathematics

Iteration method of approximate solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order. (English) Zbl 1424.34062
Summary: We construct an iteration sequence converging (in the uniform norm in the space of continuous functions) to the solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order (the weak nonlinearity means the presence of a small parameter in the nonlinear term). The sequence thus constructed is also asymptotic in the sense that the difference of its $$n$$th element from the solution of the problem is proportional to the $$(n+1)$$th power of the perturbation parameter.
##### MSC:
 34A45 Theoretical approximation of solutions to ordinary differential equations 34E15 Singular perturbations, general theory for ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:
##### References:
 [1] Barashkov AS. Small Parameter Method in Multidimensional Inverse Problems. Inverse and Ill-Posed Problems Series, Vol. 12. Utrecht, the Netherlands: VSP, 1998. 2852 ALIMOV and BUKZHALEV/Turk J Math [2] Boglaev YP. An iterative method for the approximate solution of singularly perturbed problems. Soviet Math Dokl 1976; 17: 543-547. · Zbl 0351.34007 [3] Boglaev YP, Zhdanov AV, Stel’makh VG. Uniform approximations for solutions of certain singularly perturbed nonlinear equations. Differ Equat 1978; 14: 273-281. [4] Bukzhalev EE. The Cauchy problem for singularly perturbed weakly nonlinear second-order differential equations: an iterative method. Moscow Univ Comput Math Cybern 2017; 41: 113-121. · Zbl 1394.65049 [5] Bukzhalev EE. A method of the study of singularly perturbed differential equations. Memoirs Faculty Phys Moscow State Univ 2017; 4: 1740304. [6] Bukzhalev EE. On one method for the analysis of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation. Comp Math Phys 2017; 57: 1635-1649. · Zbl 1384.34067 [7] Bukzhalev EE. A method for studying the Cauchy problem for a singularly perturbed weakly nonlinear first-order differential equation. Mosc U Phys Bull 2018; 73: 53-56. [8] Bukzhalev EE. A method to study the Cauchy problem for a singularly perturbed homogeneous linear differential equation of arbitrary order. Mosc Univ Math Bull 2018; 73: 41-49. · Zbl 1406.34049 [9] Bukzhalev EE. Uniform exponential-power estimate for the solution to a family of the Cauchy problems for linear differential equations. arXiv: 1802.09486. [10] Bukzhalev EE, Ovchinnikov AV. A method of the study of the Cauchy problem for a singularly perturbed linear inhomogeneous differential equation. Australian Journal of Mathematical Analysis and Applications 2018; 15: 1-14. [11] Gantmacher FR. The Theory of Matrices, Vol. 2. AMS Chelsea Publishing Series, Vol. 133. Providence, RI, USA: AMS Chelsea Publishing, 2000. [12] King AC, Billingham J, Otto SR. Differential Equations: Linear, Nonlinear, Ordinary, Partial. Cambridge, UK: Cambridge University Press, 2003. [13] Tikhonov AN, Vasil’eva AB, Sveshnikov AG. Differential Equations, 1st ed. Springer Series in Soviet Mathematics. Berling, Germany: Springer-Verlag, 1985. [14] Tyrtyshnikov EE. A Brief Introduction to Numerical Analysis, 1st ed.New York, NY, USA: Springer Science+Business Media, 1997. [15] Vasil’eva AB, Butuzov VF, Kalachev LV. The Boundary Function Method for Singular Perturbation Problems. SIAM Studies in Applied Mathematics. Philadelphia, PA, USA: SIAM, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.