[Paper Review] Développements asymptotiques combinés et points tournants d'équations différentielles singulièrement perturbées

Augustin Fruchard, Reinhard Schäfke|arXiv (Cornell University)|Apr 29, 2010

Differential Equations and Numerical Methods25 references3 citations

TL;DR

This paper introduces a novel class of combined asymptotic developments (dac) for singularly perturbed ordinary differential equations near turning points, where classical asymptotic methods fail. By blending slow components (related to formal solutions) and fast components (from inner layer scaling), the method provides uniform approximations across regions near and away from turning points, using Gevrey-type estimates and a complex-analytic approach inspired by Ramis-Sibuya theory.

ABSTRACT

We develop the theory of a new type of asymptotic expansions for functions of two variables the coefficients of which contain functions of one of the variables as well as functions of the quotient of these two variables. These combined asymptotic expansions (DAC) are particularly well suited for the description of solutions of singularly perturbed ordinary differential equations in the neighborhood of turning points. We describe the relations with the method of matched asymptotic expansions and with the classical composite asymptotic expansions used for boundary layers. We present a result of Ramis-Sibuya type that proves the existence of these DAC and provides Gevrey estimates. Three applications are given, two of which depend on Gevrey estimates for the DAC. In our article, we only apply the theory to scalar ordinary differential equations, but we are convinced that they will be very useful for systems of differential equations and other types of functional equations as well.

Motivation & Objective

• Address the breakdown of classical matched asymptotic expansions at turning points in singularly perturbed ODEs.
• Develop a new framework for combined asymptotic expansions (dac) that uniformly approximates solutions near turning points.
• Overcome the failure of standard formal power series due to singularities in coefficients at turning points.
• Establish a rigorous connection between Gevrey asymptotic theory and the structure of solutions near turning points.
• Provide a systematic method for constructing uniform approximations valid both in outer and inner layers using analytic continuation and sectorial estimates.

Proposed method

• Propose a new form of combined asymptotic development: $ \sum_{n \geq 0} \left( a_n(x) + g_n\left(\frac{x - x^*}{\eta}\right) \right) \eta^n $, where $ \eta = \varepsilon^{1/2} $, blending slow and fast components.
• Define $ a_n(x) $ as the regular part of the formal solution coefficients $ y_n(x) $, and $ g_n $ as the regular part at infinity of the inner equation solution.
• Use a direct method based on formal solution analysis and existence proof of analytic solutions with the given dac.
• Apply a complex-analytic approach via Ramis-Sibuya theory: construct solutions on sectorial covers of the origin with exponential estimates.
• Establish Gevrey-type estimates for both the main expansion and the fast components $ g_n $, using coherent sectorial coverings.
• Leverage Borel-Ritt-type theorems and plate functions to reconstruct solutions from formal series with Gevrey properties.

Experimental results

Research questions

• RQ1 How can combined asymptotic developments be generalized to handle turning points where classical methods fail?
• RQ2 What is the structure of the formal solution near a turning point, and how can its singularities be regularized?
• RQ3 Can a unified asymptotic expansion be constructed that is valid both in the outer region and in the inner layer near a turning point?
• RQ4 What role does Gevrey asymptotic theory play in controlling the growth of coefficients and ensuring existence of solutions?
• RQ5 How can complex-analytic techniques like sectorial covers and exponential majorants be used to prove existence of solutions with prescribed asymptotic behavior?

Key findings

• The proposed combined asymptotic developments (dac) provide a uniform approximation of solutions to singularly perturbed ODEs across regions including both the outer domain and the inner layer near a turning point.
• The method successfully handles the breakdown of classical matched asymptotic expansions due to singularities in formal solution coefficients at turning points.
• The slow part $ a_n(x) $ corresponds to the regular part of the formal solution $ \widehat{y} $, while the fast part $ g_n $ arises from the regularized behavior at infinity of the inner equation.
• Using the Ramis-Sibuya approach, the authors prove the existence of solutions with the given dac, under sectorial conditions on $ \varepsilon $ and $ x $.
• The method yields Gevrey-type estimates not only for the main expansion but also for the functions $ g_n $, ensuring robustness and control over error terms.
• The framework is shown to be essential for two of the three applications, particularly in cases involving canard solutions and resonance, where Gevrey properties are indispensable.