Variation of Parameters and the Renormalization Group Method

O’Malley, Robert E.; Kirkinis, E.

doi:10.1111/sapm.12052

Cited by 11 publications

(6 citation statements)

References 31 publications

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“…However, it can be seen as a more or less straightforward generalization of the well-known method of variation of constants for inhomogeneous linear differential equations. As discussed in [27], this method was first proposed by Lagrange in 1788 [28], almost exactly along the lines used here.…”

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confidence: 99%

Front propagation and arrival times in networks with application to neurodegenerative diseases

Putra

Oliveri

Thompson

et al. 2022

Preprint

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Many physical, epidemiological, or physiological dynamical processes on networks support front-like propagation, where an initial localized perturbation grows and systematically invades all nodes in the network. A key question is then to extract estimates for the dynamics. In particular, if a single node is seeded at a small concentration, when will other nodes reach the same initial concentration? Here, motivated by the study of toxic protein propagation in neurodegenerative diseases, we present and compare three different estimates for the arrival time in order of increasing analytical complexity: the linear arrival time, obtained by linearizing the underlying system; the Lambert time, obtained by considering the interaction of two nodes; and the nonlinear arrival time, obtained by asymptotic techniques. We use the classic Fisher-Kolmogorov-Petrovsky-Piskunov equation as a paradigm for the dynamics and show that each method provides different insight and time estimates. Further, we show that the nonlinear asymptotic method also gives an approximate solution valid in the entire domain and the correct ordering of arrival regions over large regions of parameters and initial conditions.

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confidence: 99%

Front propagation and arrival times in networks with application to neurodegenerative diseases

Putra

Oliveri

Thompson

et al. 2022

Preprint

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“…Chiba 11–13 defined the high‐order RG equation and the RG transformation to improve the error estimate. The RG method has been further developed in previous studies 14–19 …”

Section: Introductionmentioning

confidence: 99%

High‐order long‐time approximation of (N + 1)‐level systems with near‐resonance control

Geng

2021

Math Methods in App Sciences

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In this paper, we focus on high‐order approximate solutions of (N + 1)‐level systems with near‐resonance control over a long period of time. A high‐order renormalization group (RG) method with rigorous proof is developed to deal with such open quantum systems. By constructing high‐order RG equations, we obtained the high‐order long‐time RG approximate solutions of (N + 1)‐level systems in several kinds of near‐resonance cases. The numerical simulation results show that our high‐order RG method has the same accuracy as the fourth‐order Runge–Kutta method in O(1) time scale but more reliable in O()1ε time scale.

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“…Chiba in [11][12][13] defined the high-order RG equation and the RG transformation to improve error estimate. The RG method has been further developed in [14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

High-Order Approximation to Two-Level Systems with Quasiresonant Control

Wang

2020

Advances in Mathematical Physics

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In this paper, we focus on high-order approximate solutions to two-level systems with quasi-resonant control. Firstly, we develop a high-order renormalization group (RG) method for Schrödinger equations. By this method, we get the high-order RG approximate solution in both resonance case and out of resonance case directly. Secondly, we introduce a time transformation to avoid the invalid expansion and get the high-order RG approximate solution in near resonance case. Finally, some numerical simulations are presented to illustrate the effectiveness of our RG method. We aim to provide a mathematically rigorous framework for mathematicians and physicists to analyze the high-order approximate solutions of quasi-resonant control problems.

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Variation of Parameters and the Renormalization Group Method

Cited by 11 publications

References 31 publications

Front propagation and arrival times in networks with application to neurodegenerative diseases

Front propagation and arrival times in networks with application to neurodegenerative diseases

High‐order long‐time approximation of (N + 1)‐level systems with near‐resonance control

High-Order Approximation to Two-Level Systems with Quasiresonant Control

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