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The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems. (English) Zbl 1244.65113
Summary: A new computational method for solving a second-order nonlinear singularly perturbed boundary value problems is provided. In order to overcome a highly singular behavior very near to the boundary as being not easy to treat by numerical method, we adopt a coordinate transformation from an \(x\)-domain to a \(t\)-domain via a rescaling technique, which can reduce the singularity within the boundary layer. Then, we construct a Lie-group shooting method to search a missing initial condition through the finding of a suitable value of a parameter \(r\in [0,1]\). Moreover, we derive a closed-form formula to express the initial condition in terms of \(r\), which can be determined properly by an accurate matching to the right-boundary condition. Numerical examples are examined, showing that the present approach is highly efficient and accurate.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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