Title
Exact And Analytical Solutions For A Nonlinear Sigma Model
Keywords
approximate solution; auxiliary linear operator; control of error; exact solution; nonlinear sigma model
Abstract
We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error-minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.
Publication Date
7-30-2014
Publication Title
Mathematical Methods in the Applied Sciences
Volume
37
Issue
11
Number of Pages
1642-1651
Document Type
Article
Personal Identifier
scopus
DOI Link
https://doi.org/10.1002/mma.2926
Copyright Status
Unknown
Socpus ID
84903315712 (Scopus)
Source API URL
https://api.elsevier.com/content/abstract/scopus_id/84903315712
STARS Citation
Baxter, Mathew and Van Gorder, Robert A., "Exact And Analytical Solutions For A Nonlinear Sigma Model" (2014). Scopus Export 2010-2014. 7946.
https://stars.library.ucf.edu/scopus2010/7946