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**The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions.**
*(English)*
Zbl 1232.65108

Summary: We provide a Lie-group shooting method for the numerical solutions of second order nonlinear boundary value problems exhibiting multiple solutions. It aims to find all solutions as easy as possible. 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 using the technique of symmetric extension of the governing equations. The Lie-group shooting method is very effective to search unknown initial condition through a weighting factor \(r\in (0,1)\). Furthermore, the closed-form solutions are derived to calculate the unknown initial condition in terms of \(r\) in a more refined range identified. 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 \(r\).

### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |