In this paper, we propose a review of the free boundary formulation for BVPs defined on semi-infinite intervals. The main idea and theorem are illustrated, for the reader convenience, by using a class of second-order BVPs. Moreover, we are able to show the effectiveness of the proposed approach using two examples where the exact solution both for the BVPs and their free boundary formulation are available. Then, we describe the free boundary formulation for a general class of BVPs governed by an n-order differential equation. In this context, we report three problems solved using the free boundary formulation. The reported numerical results, obtained by the iterative transformation method or Keller’s second-order finite difference method, are found to be in very good agreement with those available in the literature. The last result of this research is that, in order to orient the interested reader, we provide an extensive bibliography. Of course, we may aspect further and more interesting applications of the free boundary formulation in the future.

The first contribution of this paper is the extension of the non-iterative transformation method, proposed by T\”opfer more than a century ago and defined for the numerical solution of the Blasius problem, to a Blasius problem with extended boundary conditions. This method, which makes use of the invariance of two physical parameters with respect to an extended scaling group of point transformations, allows us to solve numerically the Blasius problem with extended boundary conditions by solving a related initial value problem and then rescaling the obtained numerical solution. Therefore, our method is an initial value method. However, in this way, we cannot fix in advance the values of the physical parameters, and if we need just to compute the numerical solution for given values of the two parameters we have to define an iterative extension of the transformation method, which is the second contribution of this work.

In this paper, we define a non-iterative transformation method for boundary-layer flows of non-Newtonian fluids past a flat plate. The problem to be solved is an extended Blasius problem depending on a parameter. This method allows us to solve numerically the extended Blasius problem by solving a related initial value problem and then rescaling the obtained numerical solution. Therefore, it is a non-iterative initial value method. We find that our computed numerical results, for a wide range of the parameter involved, are in very good agreement with the data reported in the literature.

In this paper, we have defined the free boundary formulation for two extended Blasius problems. These problems are of interest in boundary layer theory and are deduced from the governing partial differential equations by using appropriate similarity variables. The computed results, for the so-called missing initial condition, are favourably compared with recent results available in the literature.

In this paper, we define a non-iterative transformation method for an Extended Blasius Problem. The original non-iterative transformation method, which is based on scaling invariance properties, was defined for the classical Blasius problem by T\”opfer in 1912. This method allows us to solve numerically a boundary value problem by solving a related initial value problem and then rescaling the obtained numerical solution. In recent years, we have seen applications of the non-iterative transformation method to several problems of interest. The obtained numerical results are improved by both a mesh refinement strategy and Richardson’s extrapolation technique. In this way, we can be confident that the computed six decimal places are correct.