Renormalization group second-order approximation for singularly perturbed nonlinear ordinary differential equations

• Verfasst von: Marciniak-Czochra, Anna [VerfasserIn]
• Verfasst von: Mikelić, Andro [VerfasserIn]
• Verfasst von: Stiehl, Thomas [VerfasserIn]
• Titel: Renormalization group second-order approximation for singularly perturbed nonlinear ordinary differential equations
• Verf.angabe: Anna Marciniak‐Czochra, Andro Mikelić, Thomas Stiehl
• Umfang: 20 S.
• Fussnoten: Gesehen am 17.12.2018
• Titel Quelle: Enthalten in: Mathematical methods in the applied sciences
• Jahr Quelle: 2018
• Band/Heft Quelle: 41(2018), 14, S. 5691-5710
• ISSN Quelle: 1099-1476
• DOI: doi:10.1002/mma.5107
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1585610593

Abstract

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ2). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva-O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.