Asymptotic Analysis and Singular Perturbation Theory

Journal of Physics: Conference Series, 2005

Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences, 1966

The result s of the preceding pape r are extend ed to the general system of n first-order differential eq uations having an irregular singu larit y of arbitrary rank at infinity. Formal solutions are explicitly constructed for th e syste m in c anonical form. Proofs of existence and uniqu e ness of solutions of integral equation s defining th e e rror are given. As an example, the case n = 2 is solved completely, and a flow chart of the transformations of this case to canonical form is included. .

Electronic Journal of Differential Equations

A structured and synthetic presentation of Vasil'eva's combined expansions is proposed. These expansions take into account the limit layer and the slow motion of solutions of a singularly perturbed differential equation. An asymptotic formula is established which gives the distance between two exponentially close solutions. An ``input-output" relation around a {it canard} solution is carried out in the case of turning points. We also study the distance between two canard values of differential equations with given parameter. We apply our study to the Liouville equation and to the splitting of energy levels in the one-dimensional steady Schr"{o}dinger equation in the double well symmetric case. The structured nature of our approach allows us to give effective symbolic algorithms.

Journal of research of the National Bureau of Standards

The method of Olver for bounding the error term in the asymptotic so lutions of a second-order equation having a n irregular singul arity at infinity is extended to the general system of n first-order equations in the case when the eige nvalu es of the lead coefficient matrix are distinct. Vector and norm bounds are given for th e difference between an actual solution vector and a partial su m of a for mal so luti on vector. Two cases are distinguished geometricall y: In one it is possible to exp ress the error vec tor by a s in gle Volterra vec tor integral equat.ion; in the other it is necessa ry to use a simultaneous pair of Volterra vector integral equatio ns. Some ne w inequaliti es for integral equations are given in an append ix .

Computers & Mathematics with Applications, 1987

Journal d'Analyse Mathématique, 2008

arXiv (Cornell University), 2014

We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter Λ: x m y ′′ −Λ 2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, specially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansion of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large Λ is not of exponential type, but of power type. In this case, Olver's theory does not give as many details as it gives in the cases m = 2. Then, we consider here the special case m = 2. We propose two different techniques to handle the problem: (i) a modification of Olver's method that replaces the role of the exponential approximations by power approximations and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.

2015

Abstract. A structured and synthetic presentation of Vasil’eva’s combined expansions is proposed. These expansions take into account the limit layer and the slow motion of solutions of a singularly perturbed differential equation. An asymptotic formula is established which gives the distance between two exponentially close solutions. An “input-output ” relation around a canard solution is carried out in the case of turning points. We also study the distance between two canard values of differential equations with given parameter. We apply our study to the Liouville equation and to the splitting of energy levels in the one-dimensional steady Schrödinger equation in the double well symmetric case. The structured nature of our approach allows us to give effective symbolic algorithms. 1.

Nonlinear Analysis-theory Methods & Applications, 1989

Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics

The method of Olver for bounding the error term in the asymptotic so lutions of a second-order equation having a n irregular singul arity at infinity is extended to the general system of n first-order equations in the case when the eige nvalu es of the lead coefficient matrix are distinct. Vector and norm bounds are given for th e difference between an actual solution vector and a partial su m of a for mal so luti on vector. Two cases are distinguished geometricall y: In one it is possible to exp ress the error vec tor by a s in gle Volterra vec tor integral equat.ion; in the other it is necessa ry to use a simultaneous pair of Volterra vector integral equatio ns. Some ne w inequaliti es for integral equations are given in an append ix .

Advances in Applied Mathematics, 1980

Journal of Differential Equations, 1985

Let f be a real valued continuously differentiable function on the real line R, i.e., f E C'(R). We assume the zero set f-' (0) is not empty. We also discard the trivial case f-'(O) = R. Consider the problem of finding a function U(X) which satisfies the scalar differential equation &U'(X) =f(u(x)), O<X<l, (1) with given mass m: s I u(x) dx = m.

2021

Asymptotic forms of solutions of half-linear ordinary differential equation ( |u′|α−1u′ )′ = α ( 1+b(t) ) |u|α−1u are investigated under a smallness condition and some signum conditions on b(t). When α = 1, our results reduce to well-known ones for linear ordinary differential equations.