Riccardo Fazio

A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite Intervals

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 Blasius problem formulated as a free boundary value problem

SummaryIn the present paper we point out that the correct way to solve the Blasius problem by numerical means is to reformulate it as free boundary value problem. In the new formulation the truncated boundary (instead of infinity) is the unknown free boundary and it has to be determined as part of the numerical solution. Taking into account the “partial” inavariance of the mathematical model at hand with respect to a stretching group we define a non-iterative transformation method. Further, we compare the improved numerical results, obtained by the method in point, with analytical and numerical ones. Moreover, the numerical results confirm that the question of accuracy depends on the value of the free boundary. Therefore, this indicates that boundary value problems with a boundary condition at infinity should be numerically reformulated as free boundary value problems.

Similarity and numerical analysis for free boundary value problems

We consider the similarity properties of nonlinear ordinary free boundary value problems, i.e., u″ = f(x,u,u′) x∊(0,s) s>0 u(0) = α; u(s) = u′(s) = 0; α≠0. By making use of group properties we show that for the two classes of problems it is possible to define a method that allows us to find the location of the free boundary s through the first numerical integration and the numerical solution by means of a second integration. Moreover, by requiring invariance of some parameter, we give an important extension of the method to solve a problem that does not belong to the two classes in point. Finally we remark that the method is self-validating.

A Similarity Approach to the Numerical Solution of Free Boundary Problems

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.

The falkneer-skan equation: Numerical solutions within group invariance theory

The iterative transformation method, defined within the framework of the group invariance theory, is applied to the numerical solution of the Falkner-Skan equation with relevant boundary conditions. In this problem a boundary condition at infinity is imposed which is not suitable for a numerical use. In order to overcome this difficulty we introduce a free boundary formulation of the problem, and we define the iterative transformation method that reducess the free boundary formulation to a sequence of initial value problems. Moreover, as far as the value of the wall shear stress is concerned we propose a numerical test of convergence. The usefulness of our approach is illustrated by considering the wall shear stress for the classical Homann and Hiemenz flows. In the Homanns case we apply the proposed numerical test of convergence, and meaningful numerical results are listed. Moreover, for both cases we compared our results with those reported in literature.

A survey on free boundary identification of the truncated boundary in numerical BVPs on infinite intervals

A free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in Fazio (SIAM J. Numer. Anal 33 (1996) 1473). We consider here a survey on recent developments related to the free boundary identification of the truncated boundary. The goals of this survey are: to recall the reasoning for a free boundary identification of the truncated boundary, to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature, and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line.

A numerical test for the existeence and uniqeness of solution of free boundery problems

The aim of this work is to introduce a numbeicaal test for the existence ferential eqinueness of solutionof free boundary problems govrned by an ordinary differential equtial. the main result is given by a theorem relation the existence and uniqueness question to thenumber of real zeal zeros of a function implicitly defined within the formulation of the iterative tranesformation method .as a consequence we can investigate the existence and uniquess of solution by studing the behaviour of that fution within such a context the numerical test is illustrated by two examples.

Normal variables transformation method applied to free boundary value problems

We extend the application of group analysis approach to determining the numerical solution of free boundary value problems. If the differential problem is invariant under a translation group of transformations we will formulate a non-iterative method of solution. This is done by introducing the concept of normal variables. Application of the method to two problems in the class characterized produces correct numerical results. Moreover, introducing a parameter into the differential problem and requiring invariance under an extended stretching group we give an iterative method applicable to any free boundary value problem. As further result of the knowledge of the group properties we point out that these methods are self-validating. Finally we suggest application of numerical transformation methods to boundary value problems.

The iterative transformation method: numerical solution of one-dimensional parabolic moving boundary problems

The main contribution of this paper is the application of the iterative transformation method to the numerical solution of the sequence of free boundary problems obtained from one-dimensional parabolic moving boundary problems via the implicit Eulers method. The combination of the two methods represents a numerical approach to the solution of those problems. Three parabolic moving boundary problems, two with explicit and one with implicit moving boundary conditions, are solved in order to test the validity of the proposed approach. As far as the moving boundary position is concerned the obtained numerical results are found to be in agreement with those available in literature.The main contribution of this paper is the application of the iterative transformation method to the numerical solution of the sequence of free boundary problems obtained from one-dimensional parabolic moving boundary problems via the implicit Eulers method. The combination of the two methods represents a numerical approach to the solution of those problems. Three parabolic moving boundary problems, two with explicit and one with implicit moving boundary conditions, are solved in order to test the validity of the proposed approach. As far as the moving boundary position is concerned the obtained numerical results are found to be in agreement with those available in literature.

Blasius problem and Falkner–Skan model: Töpfer’s algorithm and its extension

Abstract In this paper, we review the so-called Topfer algorithm that allows us to find a non-iterative numerical solution of the Blasius problem, by solving a related initial value problem and applying a scaling transformation. Moreover, we remark that the applicability of this algorithm can be extended to any given problem, provided that the governing equation and the initial conditions are invariant under a scaling group of point transformations and that the asymptotic boundary condition is non-homogeneous. Then, we describe an iterative extension of Topfer’s algorithm that can be applied to a general class of problems. Finally, we solve the Falkner–Skan model, for values of the parameter where multiple solutions are admitted, and report original numerical results, in particular data related to the famous reverse flow solutions by Stewartson. The numerical data obtained by the extended algorithm are in good agreement with those obtained in previous studies.