Journal of Applied Mathematics

A Third-Order $p$-Laplacian Boundary Value Problem Solved by an SL$\left(3,ℝ\right)$ Lie-Group Shooting Method

Chein-Shan Liu

Abstract

The boundary layer problem for power-law fluid can be recast to a third-order $p$-Laplacian boundary value problem (BVP). In this paper, we transform the third-order $p$-Laplacian into a new system which exhibits a Lie-symmetry SL$\left(3,ℝ\right)$. Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of $r\in \left[0,1\right]$. The present SL$\left(3,ℝ\right)$ Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order $p$-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4) method to obtain a quite accurate numerical solution of the $p$-Laplacian.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 497863, 13 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807916

Digital Object Identifier
doi:10.1155/2013/497863

Mathematical Reviews number (MathSciNet)
MR3039723

Zentralblatt MATH identifier
1266.35110

Citation

Liu, Chein-Shan. A Third-Order $p$ -Laplacian Boundary Value Problem Solved by an SL $\left(3,ℝ\right)$ Lie-Group Shooting Method. J. Appl. Math. 2013 (2013), Article ID 497863, 13 pages. doi:10.1155/2013/497863. https://projecteuclid.org/euclid.jam/1394807916

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