# Riccardo FazioUniversità degli Studi di Messina | UNIME · Dipartimento di Matematica, Informatica, Fisica e Scienza della Terra

Riccardo Fazio

Ph.D. Mathematics

## About

128

Publications

18,458

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1,064

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Introduction

My main research topic is the numerical solution of free, moving or interface problems. More details can be found by the interested reader within the web site mat521.unime.it/fazio
Moreover, I am developing new and original numerical methods for BVPs defined on infinite intervals. A past interest was related to the numerical solution of wave propagation and hyperbolic problems. A further past interest is the numerical solution of initial value problems especially stiff problems.

Additional affiliations

January 2006 - December 2007

December 2002 - April 2019

January 2002 - present

Education

January 1986 - December 1989

October 1980 - October 1984

## Publications

Publications (128)

In this paper, we present an implicit finite difference method for the numerical solution of the Black–Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front-fixing approach where the option price and the early exercise boundary are computed simultaneously. We study the consistency...

This paper deals with a non-standard implicit finite difference scheme that is defined on a quasi-uniform mesh for approximate solutions of the Magneto-Hydro Dynamics (MHD) boundary layer flow of an incompressible fluid past a flat plate for a wide range of the magnetic parameter. The proposed approach allows imposing the given boundary conditions...

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Ou...

This paper deals with a non-standard finite difference scheme defined on a quasi-uniform mesh for approximate solutions of the Magneto-Hydro Dynamics (MHD) boundary layer flow of an incompressible fluid past a flat plate for a wide range of the magnetic parameter. We show how to improve the obtained numerical results via a mesh refinement and a Ric...

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 Töpfer in 1912. This method allows us to solve numerically a boundary value problem by solving a related i...

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 peculiar difference between a transformation and a shooting method is that the former is conceived and for...

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 compare...

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 t...

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 t...

In this paper, we have defined and applied a non-ITM to an extended Blasius problem describing a 2D laminar boundary-layer with power-law viscosity for Newtonian fluids. For a particular value of the parameter involved, this problem reduces to the celebrated Blasius problem and in this particular case, our method reduces to the T\"opfer non-iterati...

In this paper, within scaling invariance theory, we define and apply to the numerical solution of a similarity boundary layer model an iterative transformation method. The boundary value problem to be solved depends on a parameter and is defined on a semi-infinite interval. Using our transformation method we are able to solve the problem in point f...

In this paper, we have defined and applied a non-ITM to an extended Blasius problem describing a 2D laminar boundary-layer with power-law viscosity for Newtonian fluids. For a particular value of the parameter involved, this problem reduces to the celebrated Bla-sius problem and in this particular case, our method reduces to the Töpfer non-iterativ...

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...

This book describes several applications to economy and finance. Of particular interest is an explicit finite difference scheme for solving the American put option. An appendix lists OCTAVE script file.

This work is concerned with the existence and uniqueness of boundary value problems defined on infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utm...

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 t...

In this paper, we present an implicit finite difference method for the numerical solution of the Black-Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front fixing approach where the option price and the early exercise boundary are computed simultaneously. Consistency and stabilit...

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...

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 formula...

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...

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...

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 Töpfer in 1912. This method allows us to solve numerically a boundary value problem by solving a related in...

The aim of this work is to point out that the class of free boundary problems governed by second order autonomous ordinary differential equations can be transformed to initial value problems. Interest in the numerical solution of free boundary problems arises because these are always nonlinear problems. The theoretical content of this paper is orig...

This paper deals with a non-standard finite difference scheme defined on a quasi-uniform mesh for approximate solutions of the MHD boundary layer flow of an incompressible fluid past a flat plate for a wide range of the magnetic parameter. The obtained numerical results are compared with those available in the literature. We show how to improve the...

This is a bibliography file, in bib format, for my personal bibliography to be used for citation in a LaTeX document. For the neck names see rfazio.pdf

This bibliography records publications of Riccardo Fazio. The BIB file is rfazio.bib. These files are available on internet from the URL:mat521.unime.it/fazio

In the market for financial derivatives, the most important problem is the so-called option valuation problem or in a few words: the problem of computing a fair price for a given option. Analytical solutions of American options problems are seldom available, but such derivatives of financial markets can be priced by numerical methods. For the numer...

In this paper, we present an implicit finite difference method for numerical solution of Black-Scholes model of American put options. We combine the proposed numerical method by using a front fixing approach where the option price and the early exercise boundary are computed simultaneously. Consistency and stability properties of the method are stu...

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...

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...

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 Töpfer and dates more than a century ago. Here we define a non-iterative transformation method for Blasius equation with a moving wall...

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 Töpfer and dates more than a century ago. Here we define a non-iterative transformation method for Blasius equation with a moving wall...

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. I...

The present paper deals with the numerical solution of time-fractional advection-diffusion equations involving the Caputo derivative with source term by means of an unconditionally stable implicit finite difference method on quasi-uniform grids. We use a special quasi-uniform mesh in order to improve the numerical accuracy of the classical discrete...

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 nonstandard finite difference scheme previously introduced by the authors. In particular, we show...

In this paper, we present front-fixing finite difference schemes for numerical approximation of American put options model formulated as free boundary problem.

In this paper, we undertake a mathematical and numerical study of liquid dynamics models in a horizontal capillary. In particular, we prove that the classical model is ill-posed at initial time, and we recall two different approaches in order to define a well-posed problem. Moreover, for an academic test case, we compare the numerical approximation...

For the numerical solution of the American option valuation problem, we
provide a script written in MATLAB implementing an explicit finite difference
scheme. Our main contribute is the definition of a posteriori error estimator
for the American options pricing which is based on Richardson's extrapolation
theory. This error estimator allows us to fi...

In this paper, we present a study of an a posteriori estimator for the
discretization error of a non-standard finite difference scheme applied to
boundary value problems defined on an infinite interval. In particular, we show
how Richardson's extrapolation can be used to improve the numerical solution
involving the order of accuracy and numerical s...

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 t...

In this paper we present a MATLAB version of a non-standard finite difference
scheme for the numerical solution of the perpetual American put option models
of financial markets. These models can be derived from the celebrated
Black-Scholes models letting the time goes to infinity. The considered problem
is a free boundary problem defined on a semi-...

Capillary dynamics has been and is yet an important field of research,
because of its very relevant role played as the core mechanism at the base of
many applications. In this context, we are particularly interested in the
liquid penetration inspection technique. Due to the obviously needed level of
reliability involved with such a non-destructive...

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...

A L'Analisi Numerica è la branca della Matematica che si occupa degli algoritmi e dei metodi per risolvere, in forma numerica, classi di problemi matematici. Molti algoritmi e metodi nume-rici portano il nome dei grandi matematici: da Archimede a Eulero, da Gauss a Newton. Infatti, studi di analisi numeri-ca precorrono di secoli l'invenzione degli...

I metodi numerici per i problemi ai valori iniziali trovano ap-plicazione in molte discipline delle scienze applicate. Il presente libro di testo rappresenta una introduzione a tali metodi numeri-ci e si presta a essere adottato da corsi per la laurea magistrale in matematica o da corsi per i master nelle scienze applicate. Ven-gono introdotti i co...

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...

In book II of Newton's "Principia Mathematica" of 1687 several applicative
problems are introduced and solved. There, we can find the formulation of the
first calculus of variations problem that leads to the first free boundary
problem of history. The general calculus of variations problem is concerned
with the optimal shape design for the motion o...

In this paper we report and compare the numerical results for an ocean
circulation model obtained by the classical truncated boundary formulation, the
free boundary approach and a quasi-uniform grid treatment of the problem. We
apply a shooting method to the truncated boundary formulation and finite
difference methods to both the free boundary appr...

We present a non-standard second order finite difference scheme defined on quasi-uniform grids in order to solve boundary value problems on infinite intervals. The proposed approach is an effective way to solve this kind of BVPs problems. The use of a quasi-uniform mesh allows us to forget about any diffi-
culties with statement of boundary conditi...

Numerical test for four Runge-Kutta methods: constant step size.

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 equ...

In this paper we apply a scaling invariance analysis to reduce a class of
parabolic moving boundary problems to free boundary problems governed by
ordinary differential equations. As well known free boundary problems are
always non-linear and, consequently, their numerical solution is often obtained
iteratively. Among the numerical methods, develop...

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...

In this paper, we consider the adaptive numerical solution of one-dimensional models of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: the smaller is the capillary radius, the steeper becomes the initial transitory of the meniscus location derivative, and as a con...

We study high-resolution central schemes in Lagrangian coordinates for the one-dimensional system of conservation laws describing the evolution of two gases in slab geometry separated by an interface. By using Lagrangian coordinates, the in-terface is transformed to a fixed coordinate in the computational domain and, as a consequence, the movement...

We consider the adaptive strategies applicable to a simple model describing the phase lock of two coupled oscillators. This model has been used to show an instance of failure of the 0DE45 Runge-Kutta-Felberg solver implemented within the MATLAB ODE suite, see [J. D. Skufca. Analysis still matters: a surprising instance of failure of Runge-Kutta-Fel...

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent bo...

In this paper, we present a numerical operator splitting for time integration of 3D advection-diffusion-reaction problems.
In this approach, three distinct methods of second order accuracy are proposed for solving, separately, each term involved
in the model. Numerical results, obtained for advection – reported in [Fazio and Jannelli, IAENG Int. J....

The topic of this study is an extended similarity analysis for a one-dimensional model of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity, that is the only driving force acting on it. Besides the smaller is the capillary radius the steeper becomes the initial transi...

In this paper, we report a mathematical and numerical study of liquid dynamics models in a horizontal capillary. In particular, we prove that the classical model is illposed at initial time, and we present two different approaches in order to overcome this ill-posedness. By numerical viewpoint, we apply an adaptive strategy based on an onestep one-...

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 in...

This paper is concerned with a mathematical and numerical study of liquid dynamics in a horizontal capillary. We derive a two-liquids model for the prediction of capillary dynamics. This model takes into account the effects of real phenomena: like the outside flow action, or the entrapped gas inside a closed-end capillary. Moreover, the limitations...

This paper is concerned with a mathe- matical and numerical study of the eect of gas en- trapment on liquid dynamics in a closed-end horizon- tal capillary. This problem is important in order to understand how the presence of a gas inside the cap- illary can inuence the dynamics of capillary ows and the non destructive test procedures carried out t...

This paper is concerned with adaptive numerical methods for initial value problems gov- erned by systems of ordinary dierential equations. Here we introduce a novel step selection algorithm based on the simple idea that locally all continuous functions can be suitably approximated by a straight line. Finally we present two sample numerical com- put...

In this paper, we study rst and second order positive numerical methods for the advection equation. In particular, we consider the direct dis- cretization of the model problem and comment on its superiority to the so called method of lines. More- over, we investigate the accuracy, stability and posi- tivity properties of the direct discretization....

The most important feature of numeri- cal methods based on a spectral decomposition is the best convergence rate (even innite for innitely reg- ular functions) with respect to all other methods used in dealing with the solution to most of the dierential equations. However this is true under the manda- tory condition that at each time step of the ev...

The non-iterative numerical solution of nonlinear boundary value problems is a subject of great interest. The present paper is concerned with the theory of non-iterative transformation methods (TMs). These methods are defined within group invariance theory. Here we prove the equivalence between two non-iterative TMs defined by the scaling group and...

Introduced at the end of 60’s by NASA, Probability of Detection (PoD) is becoming more and more one of the main approach in order to assess, quantitatively, the general detection capabilities of a Non Destructive Inspection process. In spite of its importance, PoD can be elaborated in a variety of ways and can lead to some misinterpretations. Aleni...

The aim of this work is to point out that, within group invariance theory, some classes of bound- ary 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) often nonlinear problems. The theoretical...

Blasius problem is the simplest nonlin-ear 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...

The non-iterative numerical solution of nonlinear boundary value problems is a subject of great interest. This paper is concerned with the the- ory of non-iterative transformation methods. These methods are defined within group invariance theory. We prove the equivalence between two non-iterative transformation methods defined by the scaling group...

In this paper, we present some positive numerical methods for the advection equation. In particular, we consider two different classes of schemes for linear advection equation, the first one based on direct discretization and one based on method of lines. By theoretical point of view, the accuracy and positivity property of numerical methods are in...

In this study we use the van der Pol model to explain a novel numerical application of scaling invariance. The model in point
is not invariant to a scaling group of transformations, but by introducing an embedding parameter we are able to recover it
from an extended model which is invariant to an extended scaling group. As well known, within a simi...

This is a mathematical and numerical study of liquid dynamics in a horizontal capillary. We present a two-liquids model which takes into account the effects of real phenomena like the outside flow dynamics. Moreover, we report on results obtained by an adaptive numerical method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)