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Technician or Assistant
01 November 2002 - 01 February 2021
Numerical analysis researcher, with a focus on numerical methods for ODEs and PDEs and mathematical modeling.
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 and prove the stability of the implicit method by fixing the values of the free boundary and of its first derivative. We improve the accuracy of the computed solution via a mesh refinement based on Richardson’s extrapolation. Comparisons with some proposed methods for the American options problem are carried out to validate the obtained numerical results and to show the efficiency of the proposed numerical method. Finally, by using an a posteriori error estimator, we find a suitable computational grid requiring that the computed solution verifies a prefixed error tolerance.
Riccardo Fazio; Alessandra Insana; Alessandra Jannelli. A Front-Fixing Implicit Finite Difference Method for the American Put Options Model. Mathematical and Computational Applications 2021, 26, 30 .
AMA StyleRiccardo Fazio, Alessandra Insana, Alessandra Jannelli. A Front-Fixing Implicit Finite Difference Method for the American Put Options Model. Mathematical and Computational Applications. 2021; 26 (2):30.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Insana; Alessandra Jannelli. 2021. "A Front-Fixing Implicit Finite Difference Method for the American Put Options Model." Mathematical and Computational Applications 26, no. 2: 30.
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 at infinity exactly. We show how to improve the obtained numerical results via a mesh refinement and a Richardson extrapolation. The obtained numerical results are favourably compared with those available in the literature.
Riccardo Fazio; Alessandra Jannelli. A Non-Standard Finite Difference Scheme for Magneto-Hydro Dynamics Boundary Layer Flows of an Incompressible Fluid Past a Flat Plate. Mathematical and Computational Applications 2021, 26, 22 .
AMA StyleRiccardo Fazio, Alessandra Jannelli. A Non-Standard Finite Difference Scheme for Magneto-Hydro Dynamics Boundary Layer Flows of an Incompressible Fluid Past a Flat Plate. Mathematical and Computational Applications. 2021; 26 (1):22.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Jannelli. 2021. "A Non-Standard Finite Difference Scheme for Magneto-Hydro Dynamics Boundary Layer Flows of an Incompressible Fluid Past a Flat Plate." Mathematical and Computational Applications 26, no. 1: 22.
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. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context, the numerical test is illustrated by two examples where we find meaningful numerical results.
Riccardo Fazio. Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method. Mathematical and Computational Applications 2021, 26, 18 .
AMA StyleRiccardo Fazio. Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method. Mathematical and Computational Applications. 2021; 26 (1):18.
Chicago/Turabian StyleRiccardo Fazio. 2021. "Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method." Mathematical and Computational Applications 26, no. 1: 18.
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. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context the numerical test is illustrated by two examples where we find meaningful numerical results.
Riccardo Fazio. Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method. 2021, 1 .
AMA StyleRiccardo Fazio. Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method. . 2021; ():1.
Chicago/Turabian StyleRiccardo Fazio. 2021. "Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method." , no. : 1.
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 formulated within scaling invariance theory. The main aim of this paper is to propose a unifying framework for numerical transformation methods. The non-iterative method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As many boundary value problems cannot be solved non-iteratively because they lack the required scaling invariance an iterative extension of the method has been developed. This iterative method provides a simple numerical test for the existence and uniqueness of solutions, as shown by this author in the case of free boundary problems [Appl. Anal., 66 (1997) pp. 89-100] and proved herewith for a wide class of boundary value problems defined on a semi-infinite interval.
Riccardo Fazio. Scaling invariance theory and numerical transformation method: A unifying framework. Applications in Engineering Science 2020, 4, 100024 .
AMA StyleRiccardo Fazio. Scaling invariance theory and numerical transformation method: A unifying framework. Applications in Engineering Science. 2020; 4 ():100024.
Chicago/Turabian StyleRiccardo Fazio. 2020. "Scaling invariance theory and numerical transformation method: A unifying framework." Applications in Engineering Science 4, no. : 100024.
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 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 first six decimal places are correct.
Riccardo Fazio. A non‐iterative transformation method for an extended Blasius problem. Mathematical Methods in the Applied Sciences 2020, 44, 1996 -2001.
AMA StyleRiccardo Fazio. A non‐iterative transformation method for an extended Blasius problem. Mathematical Methods in the Applied Sciences. 2020; 44 (2):1996-2001.
Chicago/Turabian StyleRiccardo Fazio. 2020. "A non‐iterative transformation method for an extended Blasius problem." Mathematical Methods in the Applied Sciences 44, no. 2: 1996-2001.
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) .
Riccardo Fazio. The iterative transformation method. International Journal of Non-Linear Mechanics 2019, 116, 181 -194.
AMA StyleRiccardo Fazio. The iterative transformation method. International Journal of Non-Linear Mechanics. 2019; 116 ():181-194.
Chicago/Turabian StyleRiccardo Fazio. 2019. "The iterative transformation method." International Journal of Non-Linear Mechanics 116, no. : 181-194.
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, 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.
Riccardo Fazio. The non-iterative transformation method. International Journal of Non-Linear Mechanics 2019, 114, 41 -48.
AMA StyleRiccardo Fazio. The non-iterative transformation method. International Journal of Non-Linear Mechanics. 2019; 114 ():41-48.
Chicago/Turabian StyleRiccardo Fazio. 2019. "The non-iterative transformation method." International Journal of Non-Linear Mechanics 114, no. : 41-48.
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 dudx(0)=−1.1917906497194208 calculated for α=0.5α=0.5 with good capturing the essential behavior of u(x)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.
Riccardo Fazio; Alessandra Jannelli; Tiziana Rotondo. Numerical study on gas flow through a micro–nano porous medium based on finite difference schemes on quasi-uniform grids. International Journal of Non-Linear Mechanics 2018, 105, 186 -191.
AMA StyleRiccardo Fazio, Alessandra Jannelli, Tiziana Rotondo. Numerical study on gas flow through a micro–nano porous medium based on finite difference schemes on quasi-uniform grids. International Journal of Non-Linear Mechanics. 2018; 105 ():186-191.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Jannelli; Tiziana Rotondo. 2018. "Numerical study on gas flow through a micro–nano porous medium based on finite difference schemes on quasi-uniform grids." International Journal of Non-Linear Mechanics 105, no. : 186-191.
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.
Riccardo Fazio; Alessandra Jannelli; Santa Agreste. A Finite Difference Method on Non-Uniform Meshes for Time-Fractional Advection–Diffusion Equations with a Source Term. Applied Sciences 2018, 8, 960 .
AMA StyleRiccardo Fazio, Alessandra Jannelli, Santa Agreste. A Finite Difference Method on Non-Uniform Meshes for Time-Fractional Advection–Diffusion Equations with a Source Term. Applied Sciences. 2018; 8 (6):960.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Jannelli; Santa Agreste. 2018. "A Finite Difference Method on Non-Uniform Meshes for Time-Fractional Advection–Diffusion Equations with a Source Term." Applied Sciences 8, no. 6: 960.
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 how Richardson extrapolation can be used to improve the numerical solution using the order of accuracy and numerical solutions from 2 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.
Riccardo Fazio; Alessandra Jannelli. BVPs on infinite intervals: A test problem, a nonstandard finite difference scheme and a posteriori error estimator. Mathematical Methods in the Applied Sciences 2017, 40, 6285 -6294.
AMA StyleRiccardo Fazio, Alessandra Jannelli. BVPs on infinite intervals: A test problem, a nonstandard finite difference scheme and a posteriori error estimator. Mathematical Methods in the Applied Sciences. 2017; 40 (18):6285-6294.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Jannelli. 2017. "BVPs on infinite intervals: A test problem, a nonstandard finite difference scheme and a posteriori error estimator." Mathematical Methods in the Applied Sciences 40, no. 18: 6285-6294.
In this paper, we present front-fixing finite difference schemes for numerical approximation of American put options model formulated as free boundary problem.
Riccardo Fazio; Alessandra Insana; Alessandra Jannelli. Front fixing finite difference schemes for American put options model. INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2015 (ICNAAM 2015) 2016, 1738, 480123 .
AMA StyleRiccardo Fazio, Alessandra Insana, Alessandra Jannelli. Front fixing finite difference schemes for American put options model. INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2015 (ICNAAM 2015). 2016; 1738 ():480123.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Insana; Alessandra Jannelli. 2016. "Front fixing finite difference schemes for American put options model." INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2015 (ICNAAM 2015) 1738, no. : 480123.
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.Comment: 16 pages, 4 figures, 2 table
Riccardo Fazio. A non-iterative transformation method for Blasius equation with moving wall or surface gasification. International Journal of Non-Linear Mechanics 2016, 78, 156 -159.
AMA StyleRiccardo Fazio. A non-iterative transformation method for Blasius equation with moving wall or surface gasification. International Journal of Non-Linear Mechanics. 2016; 78 ():156-159.
Chicago/Turabian StyleRiccardo Fazio. 2016. "A non-iterative transformation method for Blasius equation with moving wall or surface gasification." International Journal of Non-Linear Mechanics 78, no. : 156-159.
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 approximations, obtained by an adaptive initial value problem solver based on an one-step one-method approach, with new asymptotic solutions. This is a possible way to validate the adaptive numerical approach for its application to real liquids.
Riccardo Fazio; Alessandra Jannelli. Well-posed initial conditions and numerical methods for one-dimensional models of liquid dynamics in a horizontal capillary. Computational and Applied Mathematics 2015, 36, 903 -913.
AMA StyleRiccardo Fazio, Alessandra Jannelli. Well-posed initial conditions and numerical methods for one-dimensional models of liquid dynamics in a horizontal capillary. Computational and Applied Mathematics. 2015; 36 (2):903-913.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Jannelli. 2015. "Well-posed initial conditions and numerical methods for one-dimensional models of liquid dynamics in a horizontal capillary." Computational and Applied Mathematics 36, no. 2: 903-913.
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 solutions from two nested quasi-uniform grids. A benchmark problem is examined for which the exact solution is known and we get the following result: if the round-off error is negligible and the grids are sufficiently fine then the Richardson's error estimate gives an upper bound of the global error.
Riccardo Fazio; Alessandra Jannelli. A Posteriori Error Estimator for a Non-Standard Finite Difference Scheme Applied to BVPs on Infinite Intervals. 2015, 1 .
AMA StyleRiccardo Fazio, Alessandra Jannelli. A Posteriori Error Estimator for a Non-Standard Finite Difference Scheme Applied to BVPs on Infinite Intervals. . 2015; ():1.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Jannelli. 2015. "A Posteriori Error Estimator for a Non-Standard Finite Difference Scheme Applied to BVPs on Infinite Intervals." , no. : 1.
Riccardo Fazio. The iterative transformation method for the Sakiadis problem. Computers & Fluids 2015, 106, 196 -200.
AMA StyleRiccardo Fazio. The iterative transformation method for the Sakiadis problem. Computers & Fluids. 2015; 106 ():196-200.
Chicago/Turabian StyleRiccardo Fazio. 2015. "The iterative transformation method for the Sakiadis problem." Computers & Fluids 106, no. : 196-200.
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.
Riccardo Fazio. Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods. Acta Applicandae Mathematicae 2014, 140, 27 -42.
AMA StyleRiccardo Fazio. Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods. Acta Applicandae Mathematicae. 2014; 140 (1):27-42.
Chicago/Turabian StyleRiccardo Fazio. 2014. "Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods." Acta Applicandae Mathematicae 140, no. 1: 27-42.
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 finite differences 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 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.Comment: 22 pages, 3 figures, 7 table
Riccardo Fazio; Alessandra Jannelli. Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals. Journal of Computational and Applied Mathematics 2014, 269, 14 -23.
AMA StyleRiccardo Fazio, Alessandra Jannelli. Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals. Journal of Computational and Applied Mathematics. 2014; 269 ():14-23.
Chicago/Turabian StyleRiccardo Fazio; Alessandra Jannelli. 2014. "Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals." Journal of Computational and Applied Mathematics 269, no. : 14-23.
Riccardo Fazio. A non-iterative transformation method for Newton's free boundary problem. International Journal of Non-Linear Mechanics 2014, 59, 23 -27.
AMA StyleRiccardo Fazio. A non-iterative transformation method for Newton's free boundary problem. International Journal of Non-Linear Mechanics. 2014; 59 ():23-27.
Chicago/Turabian StyleRiccardo Fazio. 2014. "A non-iterative transformation method for Newton's free boundary problem." International Journal of Non-Linear Mechanics 59, no. : 23-27.
Riccardo Fazio; Salvatore Iacono. An analytical and numerical study of liquid dynamics in a one-dimensional capillary under entrapped gas action. Mathematical Methods in the Applied Sciences 2013, 37, 2923 -2933.
AMA StyleRiccardo Fazio, Salvatore Iacono. An analytical and numerical study of liquid dynamics in a one-dimensional capillary under entrapped gas action. Mathematical Methods in the Applied Sciences. 2013; 37 (18):2923-2933.
Chicago/Turabian StyleRiccardo Fazio; Salvatore Iacono. 2013. "An analytical and numerical study of liquid dynamics in a one-dimensional capillary under entrapped gas action." Mathematical Methods in the Applied Sciences 37, no. 18: 2923-2933.