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

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$(3,ℝ)$. 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 $r\in [0,1]$. The present SL$(3,ℝ)$ 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.