TAILIEUCHUNG - Iteration method of approximate solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order
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). | Turk J Math (2018) 42: 2841 – 2853 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Iteration method of approximate solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order 1 Alexey R. ALIMOV1,2,∗, Evgeny E. BUKZHALEV3 Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia 2 Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia 3 Faculty of Physics, Moscow State University, Moscow, Russia Received: • Accepted/Published Online: • Final Version: Abstract: 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 departure of its n th element from the solution of the problem is proportional to the (n + 1) th power of the perturbation parameter. Key words: Singular perturbations, Banach contraction principle, method of asymptotic iterations, Routh–Hurwitz stability criterion 1. Introduction Let f i : A → B, where i ∈ 1, m . Throughout, (f 1 , . . . , f m ) will denote the vector function f : A → Bm , x 7→ (f 1 (x), . . . f m (x)). ∞ In the present paper we propose a method of constructing a sequence {ψn ( · ; ε)}n=0 of functions ψn ( · ; ε) := (ψn1 ( · ; ε), . . . , ψnm ( · ; ε)), which is convergent, for any ε ∈ (0, ε0 ), in the norm of the space Cm [0, X] of m -dimensional continuous vector functions on [0, X], to a function ψ( · ; ε) := (y( · ; ε), y ′ ( · ; ε) , . . . , y (m−1) ( · ; ε)), where y( · ; ε) is the classical solution to the problem (1)–(2) (here and in what follows, by the derivative we mean the derivative .
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