The classical numerical treatment of two-point boundary value problems on infinite intervals is based on the introduction of a truncated boundary (instead of infinity) where appropriate boundary conditions are imposed. Then, the truncated boundary allowing for a satisfactory accuracy is computed by trial. Motivated by several problems of interest in boundary layer theory, here we consider boundary value problems on infinite intervals governed by a third-order ordinary differential equation. We highlight a novel approach to define the truncated boundary. The main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero. In the proposed formulation, the truncated boundary is an unknown free boundary and has to be determined as part of the solution. For the numerical solution of the free boundary formulation, a noniterative and an iterative transformation method are introduced. Furthermore, we characterize the class of free boundary value problems that can be solved noniteratively. A nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved. Moreover, the Falkner–Skan equation with relevant boundary conditions is considered and representative results, obtained by the iterative transformation method, are listed for the Homann flow. All the obtained numerical results clearly indicate the effectiveness of our approach. Finally, we discuss the possible extensions of the proposed approach and for the question of a priori error analysis.

The present paper deals with the numerical solution of time-fractional advection–diffusion equations involving the Caputo derivative with a source term by means of an unconditionally-stable, implicit, finite difference method on non-uniform grids. We use a special non-uniform mesh in order to improve the numerical accuracy of the classical discrete fractional formula for the Caputo derivative. The stability and the convergence of the method are discussed. The error estimates established for a non-uniform grid and a uniform one are reported, to support the theoretical results. Numerical experiments are carried out to demonstrate the effectiveness of the method.

The aim of this work is to point out that within a similarity approach some classes of free boundary value problems governed by ordinary differential equations can be transformed to initial value problems. The interest in the numerical solution of free boundary problems arises because these are always nonlinear problems. Furthermore we show that free boundary problems arise also via a similarity analysis of moving boundary hyperbolic problems and they can be obtained as approximations of boundary value problems defined on infinite intervals. Most of the theoretical content of this survey is original: it generalizes and unifies results already available in literature. As far as applications of the proposed approach are concerned, three problems of interest are considered and numerical results for each of them are reported.