What is the best numerical method for solving ODEs in the half line?

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A. H. Bhrawy

The spectral collocation method based on rational functions.

Faraz Masoumi

Shooting method,

Hammad Khalil

Dear A. H. Bhrawy,

I think you may better know about it . But if we implement Laguerre polynomials together with operational matrices of integration and differentiation we will easily convert the problem to system of algebric equations.