Open Access

ARTICLE

The Lie-group Shooting Method for Radial Symmetric Solutions of the Yamabe Equation

S. Abbasbandy^1,2, R.A. Van Gorder³, M. Hajiketabi¹
1 Department of Mathematics, Imam Khomeini International University, Ghazvin 34149, Iran.
2 Corresponding author. Tel: +98-912-1305326; Fax: +98-283-3901312; E-mail: abbasbandy@yahoo.com (S. Abbasbandy)
3 Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe ObservatoryQuarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom

Computer Modeling in Engineering & Sciences 2015, 104(4), 329-351. https://doi.org/10.3970/cmes.2015.104.329

Abstract

We transform the Yamabe equation on a ball of arbitrary dimension greater than two into a nonlinear singularly boundary value problem on the unit interval [0,1]. Then we apply Lie-group shooting method (LGSM) to search a missing initial condition of slope through a weighting factor r ∈ (0,1). The best r is determined by matching the right-end boundary condition. When the initial slope is available we can apply the group preserving scheme (GPS) to calculate the solution, which is highly accurate. By LGSM we obtain precise radial symmetric solutions of the Yamabe equation. These results are useful in demonstrating the utility of Lie-group based numerical approaches to solving nonlinear differential equations.

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Keywords

Yamabe equation, nonlinear singularly boundary value problem, group preserving scheme, Lie-group shooting method.

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