The Lie-Group Shooting Method for Nonlinear Two-Point Boundary Value Problems Exhibiting Multiple Solutions

The present paper provides a Lie-group shooting method for the numerical solutions of second ordernonlinearboundaryvalueproblemsexhibitingmul- tiplesolutions. It aims to find all solutionsas easy as pos- sible. The boundary conditions considered are classified into four types, namely the Dirichlet, the first Robin, the second Robin and the Neumann. The two Robin type problems are transformed into a canonical one by us- ing the technique of symmetric extension of the govern- ing equations. The Lie-group shooting method is very effective to search unknown initial condition through a weighting factor r ∈ (0,1). Furthermore, the closed- form solutions are derived to calculate the unknown ini- tial condition in terms of r in a more refined range iden- tified. Numerical examples were examined to show that the new approach is highly efficient and accurate. The number of solutions can be identified in advance, and all possible solutions can be integrated readily through the obtained initial conditions by selecting suitable r.