Developing an Lie-group shooting method for a singular ϕ-Laplacian in a nonlinear ODE

In the present paper, we develop an SL ( 2 , R ) Lie-group shooting method for a regular or singular ϕ-Laplacian in a nonlinear ordinary differential equation (ODE). By using the closure property of the Lie-group, one-step Lie-group transformation between the boundary values at two ends is established. Hence, we can derive a closed-form formula in terms of r ∈ [ 0 , 1 ] to determine the missing left-boundary condition by matching the right-boundary condition through a few iterations. The present method is easy to numerical implementation with a cheap computational cost. Numerical examples are examined to show that the present SL ( 2 , R ) Lie-group shooting method is effective to treat the regular and singular ϕ-Laplacian with a high accuracy.