Riccardo Fazio Research Web-Page

Numerical Methods for Boundary Value Problems defined on Infinite Intervals

Selected List of Papers

The Blasius Problem Formulated as a Free Boundary Value Problem , Riccardo Fazio, Acta Mech., 95 (1992) 1-7.

Abstract: In the present paper we point out that the correct way to solve the Blasius problem by numerical means is to reformulate it as a 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" invariance 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.


The Falkner-Skan Equation: Numerical Solutions within Group Invariance Theory, Riccardo Fazio, Calcolo, 31 (1994) 115-124.

Abstract: 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 reduces the free boundary formulation to a sequences 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 Homann's case we apply the proposed numerical test of convergence, and meaningful numerical results are listed. Moreover, for both cases we compare our results with those reported in literature.


A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite Intervals, Riccardo Fazio, SIAM J. Numer. Anal., 33 (1996) 1473--1483.

Abstract: 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 in order 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 non-iterative and an iterative transformation method are introduced. Furthermore, we characterize the class of free boundary value problems that can be solved non-iteratively. 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 the proposed approach. Finally, we discuss some open problems and possible extensions for our approach.


A Survey on Free Boundary Identification of the Truncated Boundary in Numerical BVPs on Infinite Intervals, Riccardo Fazio, J. Comp. Appl. Math. 140 (2002) 331--344.

Abstract: A free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in [R. Fazio, SIAM J. Numer. Anal., 33 (1996) 1473-1483]. 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 Free Boundary Approach and Keller's Box Scheme for Boundary Value Problems on Infinite Intervals, Riccardo Fazio, J. Comp. Appl. Math. 80, (2003) 1549--1560.

Abstract: A free boundary approach for the numerical solution of boundary value problems (BVPs) governed by a third-order differential equation and defined on infinite intervals was proposed recently [SIAM J. Numer. Anal., 33 (1996), pp. 1473–1483]. In that approach, the free boundary (that can be considered as the truncated boundary) is unknown and has to be found as part of the solution. This eliminates the uncertainty related to the choice of the truncated boundary in the classical treatment of BVPs defined on infinite intervals. In this article, we investigate some open questions related to the free boundary approach. We recall the extension of that approach to problems governed by a system of first-order differential equations, and for the solution of the related free boundary problem we consider now the reliable Keller’s box difference scheme. Moreover, by solving a challenging test problem of interest in foundation engineering, we verify that the proposed approach is applicable to problems where none of the solution components is a monotone function.


Numerical Transformation Methods: Blasius Problem and its Variants, Riccardo Fazio, Appl. Math. Comput., 215, (2009) 1513--1521.

Abstract: Blasius problem is the simplest nonlinear boundary-layer problem. We hope that any approach developed for this epitome can be extended to more difficult hydrodynamics problems. With this motivation we review the so called Töpfer transformation, which 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. The applicability of a non-iterative transformation method to the Blasius problem is a consequence of its partial invariance with respect to a scaling group. Several problems in boundary-layer theory lack this kind of invariance and cannot be solved by non-iterative transformation methods. To overcome this drawback, we can modify the problem under study by introducing a numerical parameter, and require the invariance of the modified problem with respect to an extended scaling group involving this parameter. Then we apply initial value methods to the most recent developments involving variants and extensions of the Blasius problem.


Blasius Problem and Falkner-Skan model: Toepfer's Algorithm and its Extension, Riccardo Fazio, Computers Fluids, 73 (2013) 202--209.

Abstract: In this paper, we review the so-called Töpfer 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 Töpfer’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.


Finite Difference Schemes on Quasi-Uniform Grids for BVPs on Infinite Intervals., Riccardo Fazio and Alessandra Jannelli, J. Comput. Appl. Math., 269 (2014) 14--23.

Abstract: The classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary. A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach. On the other hand, the free boundary approach overcomes the need for a priori definition of the truncated boundary. In fact, in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution. In this paper we consider a different way to overcome the introduction of a truncated boundary, namely non-standard finite difference schemes defined on quasi-uniform grids. A quasi-uniform grid allows us to describe the infinite domain by a finite number of intervals. The last node of such grid is placed on infinity so that right boundary conditions are taken into account exactly. We apply the proposed approach to the Falkner-Skan model and to a problem of interest in foundation engineering. The obtained numerical results are found in good agreement with those available in literature. Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson’s extrapolation. Finally, we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line.


Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods, Riccardo Fazio, Acta Appl. Math., 140 (2015) 27--42.

Abstract: This paper is concerned with two examples on the application of the free boundary formulation to BVPs on a semi-infinite interval. In both cases we are able to provide the exact solution of both the BVP and its free boundary formulation. Therefore, these problems can be used as benchmarks for the numerical methods applied to BVPs on a semi-infinite interval and to free BVPs. Moreover, we emphasize how for two classes of free BVPs, we can define non-iterative initial value methods, whereas BVPs are usually solved iteratively. These non-iterative methods can be deduced within Lie’s group invariance theory. Then, we show how to apply the non-iterative methods to the two introduced free boundary formulations in order to obtain meaningful numerical results. Finally, we indicate several problems from the literature where our non-iterative transformation methods can be applied.


The Iterative Transformation Method for the Sakiadis Problem, Riccardo Fazio, Computers Fluids, 106 (2015) 196--200.

Abstract: In a transformation method the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. This paper is concerned with the application of the iterative transformation method to the Sakiadis problem. This method is an extension of the Toepfer's non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As shown by this author [Appl. Anal., 66 (1997) pp. 89-100] the method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton's root-finder. The obtained numerical results compare well with those available in literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner-Skan model [Comput. Fluids, 73 (2013) pp. 202-209].


A Non-Iterative Transformation Method for Blasius Equation with Moving Wall or Surface Gasification, Int. J. Non-Linear Mech., 78, (2016) 156--159.

Abstract: We define a non-iterative transformation method for Blasius equation with moving wall or surface gasification. The defined method allows us to deal with classes of problems in boundary layer theory that, depending on a parameter, admit multiple or no solutions. This approach is particularly convenient when the main interest is on the behaviour of the considered models with respect to the involved parameter. The obtained numerical results are found to be in good agreement with those available in literature.


BVPs on infinite intervals: a test problem, a non-standard finite difference scheme and a posteriori error estimator, Riccardo Fazio and Alessandra Jannelli, Math. Meth. Appl. Sci., 40 (2017) 6285--6294.

Abstract: As far as the numerical solution of boundary value problems defined on an infinite interval is concerned, in this paper, we present a test problem for which the exact solution is known. Then we study an a posteriori estimator for the global error of a non-standard finite difference scheme previously introduced by the authors. In particular, we show how Richardson extrapolation can be used to improve the numerical solution using the order of accuracy and numerical solutions from two nested quasi-uniform grids. We observe that if the grids are sufficiently fine the Richardson error estimate gives an upper bound of the global error.


Numerical study on gas flow through a micro-nano porous medium based on finite difference schemes on quasi-uniform grids, Riccardo Fazio, Alessandra Jannelli and Tiziana Rotondo Int. J. Non-Linear Mech., 105, (2018) 186--191.

Abstract: In this paper, the unsteady isothermal flow of a gas through a semi-infinite micro–nano porous medium described by a non-linear two-point boundary value problem on a semi-infinite interval has been considered. We solve this problem by a nonstandard finite difference method defined on quasi-uniform grids in order to derive a new numerical approximation. By introducing a stencil that is constructed in such a way that the boundary conditions at infinity are exactly assigned, the proposed method is effectively used to determine the numerical solution. In addition, a mesh refinement and the Richardson’s extrapolation allow to improve the accuracy of the numerical solution and to define a posteriori estimator for the global error of the proposed numerical scheme. We determine the accurate initial slope du/dx(0) = -1.1917906497194208 calculated for alpha = 0.5 with good capturing the essential behavior of u(x). This clearly demonstrates that the numerical solutions presented in this paper result highly accurate and in excellent agreement with the existing solutions available in the literature.


Two finite difference methods for a nonlinear BVP arising in physical oceanography, Riccardo Fazio and Alessandra Jannelli, AAPP | Atti della Accademia Peloritana dei Pericolanti, 96, A3, (2018).

Abstract: In this paper we define two finite difference methods for a nonlinear boundary value problem on infinite interval. In particular, we report and compare the numerical results for an ocean circulation model obtained by the free boundary approach and a treatment of the problem on the original semi-infinite domain by introducing a quasi-uniform grid. In the first case we apply finite difference formulae on a uniform grid and in the second case we use non-standard finite differences on a quasi-uniform grid. We point out how both approaches represent reliable ways to solve boundary value problems defined on semi-infinite intervals. In fact, both approaches overcome the need to define a priori, or find by trials, a suitable truncated boundary used by the classical numerical treatment of boundary value problems defined on a semi-infinite interval. Finally, the reported numerical results allow to point out how the finite difference method with a quasi-uniform grid is the least demanding approach between the two and that the free boundary approach provides a more reliable formulation than the classical truncated boundary one.


The Non-Iterative Transformation Method, Riccardo Fazio, Int. J. Non-Linear Mech., 114 (2019) 41--48.

Abstract: The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semi-infinite flat plate. The definition of a non-iterative transformation method for the celebrated Blasius problem is due to T{\"o}pfer and dates more than a century ago. Here we define a non-iterative transformation method for Blasius equation with a moving wall, a slip flow condition or a surface gasification. The defined method allows us to deal with classes of problems in boundary layer theory that, depending on a parameter, admit multiple or no solutions. This approach is particularly convenient when the main interest is on the behaviour of the considered models with respect to the involved parameter. The obtained numerical results are found to be in good agreement with those available in literature.


The Iterative Transformation Method, Riccardo Fazio, Int. J. Non-Linear Mech., 116 (2019) 181--194.

Abstract: In a transformation method, the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. Therefore, a transformation method, like a shooting method, is an initial value method. The main difference between a transformation and a shooting method is that the former is conceived and derive its formulation from the scaling invariance theory. This paper is concerned with the application of the iterative transformation method to several problems in the boundary layer theory. The iterative method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. This iterative method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton’s root-finder. The obtained numerical results compare well with those available in the literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner–Skan model (Fazio, 2013).


A Non-Iterative Transformation Method for an Extended Blasius Problem, Riccardo Fazio, Preprint (2020).

Abstract: 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 place are correct.

(c) by Riccardo Fazio Last modified: February 12, 2020