Nonlinear Model Equations and Variational Principles

Debnath, Lokenath

doi:10.1007/978-0-8176-8265-1_2

Nonlinear Partial Differential Equations for Scientists and Engineers 2012

DOI: 10.1007/978-0-8176-8265-1_2

|View full text |Cite

Nonlinear Model Equations and Variational Principles

Lokenath Debnath¹

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2017

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

References 92 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Slowly varying, macroscale models emerge from microscale dynamics over multiscale domains

Roberts

Bunder

2017

IMA Journal of Applied Mathematics

View full text Add to dashboard Cite

Many physical systems are well described on domains which are relatively large in some directions but relatively thin in other directions. In this scenario we typically expect the system to have emergent structures that vary slowly over the large dimensions. For practical mathematical modelling of such systems we require efficient and accurate methodologies for reducing the dimension of the original system and extracting the emergent dynamics. Common mathematical approximations for determining the emergent dynamics often rely on self-consistency arguments or limits as the aspect ratio of the 'large' and 'thin' dimensions becomes unphysically infinite. Here we build on a new approach, previously establish for systems which are large in only one dimension, which analyses the dynamics at each cross-section of the domain with a rigorous multivariate Taylor series. Then centre manifold theory supports the local modelling of the system's emergent dynamics with coupling to neighbouring cross-sections treated as a non-autonomous forcing. The union over all cross-sections then provides powerful support for the existence and emergence of a centre manifold model global in the large finite domain. Quantitative error estimates are determined from the interactions between the cross-section coupling and both fast and slow dynamics. Two examples provide practical details of our methodology. The approach developed here may be used to quantify the accuracy of known approximations, to extend such approximations to mixed order modelling, and to open previously intractable modelling issues to new tools and insights.

show abstract

Slowly varying, macroscale models emerge from microscale dynamics over multiscale domains

Roberts

Bunder

2017

IMA Journal of Applied Mathematics

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Nonlinear Model Equations and Variational Principles

Cited by 1 publication

References 92 publications

Slowly varying, macroscale models emerge from microscale dynamics over multiscale domains

Slowly varying, macroscale models emerge from microscale dynamics over multiscale domains

Contact Info

Product

Resources

About