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

Socpus ID

84903315712 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84903315712

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