A noniterative transformation method applied to two-point boundary-value problems

Abstract

Considering similarity properties of nonlinear two-point boundary-value problems with separated Dirichlet, Neumann, or Dirichlet-Neumann boundary conditions, it is possible to define a noniterative transformation method. We characterize classes of problems for which the method is applicable. Essentially we require invariance under the most general stretching or spiral group of transformation. The possibility of application of the method to a wide class problems of physical interest is shown, and some examples are treated. The main feature of the method is the possibility of determining the numerical solution after two integrations. Moreover the procedure is self-validating: the numerical solution, in order to be acceptable, has to give a good approximation of the final boundary value. We also make an extension of the method to free-boundary-value problems.