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Alessandra Jannelli
MIFT Department, University of Messina, I-98166 Messina, Italy

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Numerical analysis researcher.

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Journal article
Published: 02 August 2021 in Sensors
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In this study, an accurate analytic semi-linear elliptic differential model for a circular membrane MEMS device, which considers the effect of the fringing field on the membrane curvature recovering, is presented. A novel algebraic condition, related to the membrane electromechanical properties, able to govern the uniqueness of the solution, is also demonstrated. Numerical results for the membrane profile, obtained by using the Shooting techniques, the Keller–Box scheme, and the III/IV Stage Lobatto IIIa formulas, have been carried out, and their performances have been compared. The convergence conditions, and the possible presence of ghost solutions, have been evaluated and discussed. Finally, a practical criterion for choosing the membrane material as a function of the MEMS specific application is presented.

ACS Style

Mario Versaci; Alessandra Jannelli; Francesco Morabito; Giovanni Angiulli. A Semi-Linear Elliptic Model for a Circular Membrane MEMS Device Considering the Effect of the Fringing Field. Sensors 2021, 21, 5237 .

AMA Style

Mario Versaci, Alessandra Jannelli, Francesco Morabito, Giovanni Angiulli. A Semi-Linear Elliptic Model for a Circular Membrane MEMS Device Considering the Effect of the Fringing Field. Sensors. 2021; 21 (15):5237.

Chicago/Turabian Style

Mario Versaci; Alessandra Jannelli; Francesco Morabito; Giovanni Angiulli. 2021. "A Semi-Linear Elliptic Model for a Circular Membrane MEMS Device Considering the Effect of the Fringing Field." Sensors 21, no. 15: 5237.

Original paper
Published: 23 July 2021 in Nonlinear Dynamics
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In this paper, a two-dimensional time-fractional diffusion-reaction equation involving the Riemann–Liouville derivative is considered. Exact and numerical solutions are obtained by applying a procedure that combines the Lie symmetry analysis with the numerical methods. Lie symmetries are determined; then through the Lie transformations, the target equation is reduced into a new one-dimensional time-fractional differential equation. By solving the reduced fractional partial differential equation, exact and numerical solutions are found. The numerical solutions are determined by introducing the Caputo definition fractional derivative and by using an implicit classical numerical method. Comparisons between the numerical and exact solutions are performed.

ACS Style

Alessandra Jannelli; Maria Paola Speciale. Exact and numerical solutions of two-dimensional time-fractional diffusion–reaction equations through the Lie symmetries. Nonlinear Dynamics 2021, 105, 2375 -2385.

AMA Style

Alessandra Jannelli, Maria Paola Speciale. Exact and numerical solutions of two-dimensional time-fractional diffusion–reaction equations through the Lie symmetries. Nonlinear Dynamics. 2021; 105 (3):2375-2385.

Chicago/Turabian Style

Alessandra Jannelli; Maria Paola Speciale. 2021. "Exact and numerical solutions of two-dimensional time-fractional diffusion–reaction equations through the Lie symmetries." Nonlinear Dynamics 105, no. 3: 2375-2385.

Journal article
Published: 13 April 2021 in Mathematical and Computational Applications
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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.

ACS Style

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 Style

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 (2):30.

Chicago/Turabian Style

Riccardo 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.

Journal article
Published: 09 March 2021 in Mathematical and Computational Applications
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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.

ACS Style

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 Style

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 (1):22.

Chicago/Turabian Style

Riccardo 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.

Preprint
Published: 03 February 2021
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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 Richardson extrapolation. The obtained numerical results are favourably compared with those available in the literature.

ACS Style

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. 2021, 1 .

AMA Style

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. . 2021; ():1.

Chicago/Turabian Style

Riccardo 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." , no. : 1.

Journal article
Published: 01 January 2021 in AIMS Mathematics
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ACS Style

Alessandra Jannelli; Maria Paola Speciale. On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations. AIMS Mathematics 2021, 6, 9109 -9125.

AMA Style

Alessandra Jannelli, Maria Paola Speciale. On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations. AIMS Mathematics. 2021; 6 (8):9109-9125.

Chicago/Turabian Style

Alessandra Jannelli; Maria Paola Speciale. 2021. "On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations." AIMS Mathematics 6, no. 8: 9109-9125.

Journal article
Published: 17 September 2020 in Journal of Computational Science
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In this paper, a novel adaptive procedure for step size selection for fractional differential equations is presented. The new adaptive approach is based on the implementation of a single numerical method and uses two numerical approximations, obtained at two successive steps, to advance the computation. We define a step size selection function that allows to adapt the size of the step according to the behaviour of solution. The new approach is easy to implement and leads to a low computational cost compared to classic step doubling procedure. The reported numerical results are satisfactory and show that our adaptive approach attains more accurate results than the results obtained on uniform grids, and results as good as the step doubling procedure but with very low implementation and computational effort.

ACS Style

Alessandra Jannelli. A novel adaptive procedure for solving fractional differential equations. Journal of Computational Science 2020, 47, 101220 .

AMA Style

Alessandra Jannelli. A novel adaptive procedure for solving fractional differential equations. Journal of Computational Science. 2020; 47 ():101220.

Chicago/Turabian Style

Alessandra Jannelli. 2020. "A novel adaptive procedure for solving fractional differential equations." Journal of Computational Science 47, no. : 101220.

Journal article
Published: 09 July 2020 in IEEE Access
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In this work, numerical techniques based on Shooting procedure, Relaxation scheme and Collocation technique have been used for recovering the profile of the membrane of a $1D$ electrostatic Micro-Electro-Mechanical-Systems (MEMS) device whose analytic model considers $|\mathbf {E}|$ proportional to the membrane curvature. The comparison among these numerical techniques has put in evidence the pros and cons of each numerical procedure. Furthermore, useful convergence conditions which ensure the absence of ghost solutions, and a new condition of existence and uniqueness for the solution of the considered differential MEMS model, are obtained and discussed.

ACS Style

Mario Versaci; Alessandra Jannelli; Giovanni Angiulli. Electrostatic Micro-Electro-Mechanical-Systems (MEMS) Devices: A Comparison Among Numerical Techniques for Recovering the Membrane Profile. IEEE Access 2020, 8, 125874 -125886.

AMA Style

Mario Versaci, Alessandra Jannelli, Giovanni Angiulli. Electrostatic Micro-Electro-Mechanical-Systems (MEMS) Devices: A Comparison Among Numerical Techniques for Recovering the Membrane Profile. IEEE Access. 2020; 8 ():125874-125886.

Chicago/Turabian Style

Mario Versaci; Alessandra Jannelli; Giovanni Angiulli. 2020. "Electrostatic Micro-Electro-Mechanical-Systems (MEMS) Devices: A Comparison Among Numerical Techniques for Recovering the Membrane Profile." IEEE Access 8, no. : 125874-125886.

Journal article
Published: 01 April 2020 in Mathematics
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In this paper, a stable numerical approach for recovering the membrane profile of a 2D Micro-Electric-Mechanical-Systems (MEMS) is presented. Starting from a well-known 2D nonlinear second-order differential model for electrostatic circular membrane MEMS, where the amplitude of the electrostatic field is considered proportional to the mean curvature of the membrane, a collocation procedure, based on the three-stage Lobatto formula, is derived. The convergence is studied, thus obtaining the parameters operative ranges determining the areas of applicability of the device under analysis.

ACS Style

Mario Versaci; Giovanni Angiulli; Alessandra Jannelli. Recovering of the Membrane Profile of an Electrostatic Circular MEMS by a Three-Stage Lobatto Procedure: A Convergence Analysis in the Absence of Ghost Solutions. Mathematics 2020, 8, 487 .

AMA Style

Mario Versaci, Giovanni Angiulli, Alessandra Jannelli. Recovering of the Membrane Profile of an Electrostatic Circular MEMS by a Three-Stage Lobatto Procedure: A Convergence Analysis in the Absence of Ghost Solutions. Mathematics. 2020; 8 (4):487.

Chicago/Turabian Style

Mario Versaci; Giovanni Angiulli; Alessandra Jannelli. 2020. "Recovering of the Membrane Profile of an Electrostatic Circular MEMS by a Three-Stage Lobatto Procedure: A Convergence Analysis in the Absence of Ghost Solutions." Mathematics 8, no. 4: 487.

Journal article
Published: 08 February 2020 in Mathematics
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This paper deals with the numerical solutions of a class of fractional mathematical models arising in engineering sciences governed by time-fractional advection-diffusion-reaction (TF–ADR) equations, involving the Caputo derivative. In particular, we are interested in the models that link chemical and hydrodynamic processes. The aim of this paper is to propose a simple and robust implicit unconditionally stable finite difference method for solving the TF–ADR equations. The numerical results show that the proposed method is efficient, reliable and easy to implement from a computational viewpoint and can be employed for engineering sciences problems.

ACS Style

Alessandra Jannelli. Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences. Mathematics 2020, 8, 215 .

AMA Style

Alessandra Jannelli. Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences. Mathematics. 2020; 8 (2):215.

Chicago/Turabian Style

Alessandra Jannelli. 2020. "Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences." Mathematics 8, no. 2: 215.

Journal article
Published: 23 January 2020 in Applied Numerical Mathematics
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In this paper, numerical solutions of space-fractional advection-diffusion equations, involving the Riemann-Liouville derivative with a nonlinear source term, are presented. We propose a procedure that combines the fractional Lie symmetries analysis, to reduce the original fractional partial differential equations into fractional ordinary differential equations, with a numerical method. By adopting the Caputo definition of derivative, the reduced fractional ordinary equations are solved by applying the implicit trapezoidal method. The numerical results confirm the applicability and the efficiency of the proposed approach.

ACS Style

Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale. Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term. Applied Numerical Mathematics 2020, 155, 93 -102.

AMA Style

Alessandra Jannelli, Marianna Ruggieri, Maria Paola Speciale. Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term. Applied Numerical Mathematics. 2020; 155 ():93-102.

Chicago/Turabian Style

Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale. 2020. "Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term." Applied Numerical Mathematics 155, no. : 93-102.

Journal article
Published: 05 November 2018 in International Journal of Non-Linear Mechanics
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In this paper, the authors present a new condition of the uniqueness of the solution for a previous 1D semi-linear elliptic boundary value problem of membrane MEMS devices, where the amplitude of the electric field is considered proportional to the curvature of the membrane. The existence of the solution (membrane deflection) depends on the material of the membrane, which is obtained by Shauder-Tychonoff’s fixed point approach. Thus, in this paper, the result of uniqueness has been completely reformulated to obtain a condition depending on the material of the membrane achieving a new result of existence and uniqueness, depending on both the material of the membrane and the geometrical characteristics of the device. Then, by shooting numerical method, more realistic conditions for detecting eventual ghost solutions and new ranges of both operational parameters and mechanical tension of the membrane ensuring convergence have been achieved confirming the useful information on the industrial applicability of the model under study.

ACS Style

Mario Versaci; Giovanni Angiulli; Luisa Fattorusso; Alessandra Jannelli. On the uniqueness of the solution for a semi-linear elliptic boundary value problem of the membrane MEMS device for reconstructing the membrane profile in absence of ghost solutions. International Journal of Non-Linear Mechanics 2018, 109, 24 -31.

AMA Style

Mario Versaci, Giovanni Angiulli, Luisa Fattorusso, Alessandra Jannelli. On the uniqueness of the solution for a semi-linear elliptic boundary value problem of the membrane MEMS device for reconstructing the membrane profile in absence of ghost solutions. International Journal of Non-Linear Mechanics. 2018; 109 ():24-31.

Chicago/Turabian Style

Mario Versaci; Giovanni Angiulli; Luisa Fattorusso; Alessandra Jannelli. 2018. "On the uniqueness of the solution for a semi-linear elliptic boundary value problem of the membrane MEMS device for reconstructing the membrane profile in absence of ghost solutions." International Journal of Non-Linear Mechanics 109, no. : 24-31.

Journal article
Published: 12 October 2018 in Communications in Nonlinear Science and Numerical Simulation
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In this paper, analytical and numerical solutions of time and space fractional advection-diffusion-reaction equations are found. In general, by using Lie transformations, it is possible to reduce the fractional partial differential equations into fractional ordinary differential equations, if the symmetries admitted by target equations allow to determine the Lie transformations. In the case of the time and space fractional advection–diffusion–reaction model, the Lie symmetries do not lead to reduce the equation into fractional ordinary one. So we propose an alternative strategy to find the analytical and numerical solutions starting from the analytical and numerical results recently obtained by the authors for the time fractional advection-diffusion-reaction equation and for the space fractional advection–diffusion–reaction equation, separately, by using the Lie symmetries. The numerical results prove the efficiency and the applicability of the proposed procedure that results to be, for its high precision, a good tool to find solutions of a wide class of problems involving the fractional differential equations.

ACS Style

Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale. Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation. Communications in Nonlinear Science and Numerical Simulation 2018, 70, 89 -101.

AMA Style

Alessandra Jannelli, Marianna Ruggieri, Maria Paola Speciale. Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation. Communications in Nonlinear Science and Numerical Simulation. 2018; 70 ():89-101.

Chicago/Turabian Style

Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale. 2018. "Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation." Communications in Nonlinear Science and Numerical Simulation 70, no. : 89-101.

Journal article
Published: 01 October 2018 in International Journal of Non-Linear Mechanics
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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.

ACS Style

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 Style

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.

Chicago/Turabian Style

Riccardo 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.

Journal article
Published: 12 June 2018 in Applied Sciences
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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.

ACS Style

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 Style

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 (6):960.

Chicago/Turabian Style

Riccardo 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.

Conference paper
Published: 21 July 2017 in INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016)
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In this paper, the case of an equation involving fractional derivatives with respect to a single independent variable has been analyzed. Our aim is to determine its Lie’s symmetry, and by using them, obtain analytical and numerical solutions.

ACS Style

Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale. Analytical and numerical solutions of fractional type advection-diffusion equation. INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016) 2017, 1863, 530005 .

AMA Style

Alessandra Jannelli, Marianna Ruggieri, Maria Paola Speciale. Analytical and numerical solutions of fractional type advection-diffusion equation. INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016). 2017; 1863 (1):530005.

Chicago/Turabian Style

Alessandra Jannelli; Marianna Ruggieri; Maria Paola Speciale. 2017. "Analytical and numerical solutions of fractional type advection-diffusion equation." INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016) 1863, no. 1: 530005.

Article
Published: 27 June 2017 in Mathematical Methods in the Applied Sciences
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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.

ACS Style

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 Style

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 (18):6285-6294.

Chicago/Turabian Style

Riccardo 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.

Journal article
Published: 19 August 2015 in Computational and Applied Mathematics
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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.

ACS Style

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 Style

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 (2):903-913.

Chicago/Turabian Style

Riccardo 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.

Journal article
Published: 01 March 2006 in Applied Mathematics and Computation
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ACS Style

Riccardo Fazio; Alessandra Jannelli. Mathematical and numerical modeling for a bio-chemical aquarium. Applied Mathematics and Computation 2006, 174, 1370 -1383.

AMA Style

Riccardo Fazio, Alessandra Jannelli. Mathematical and numerical modeling for a bio-chemical aquarium. Applied Mathematics and Computation. 2006; 174 (2):1370-1383.

Chicago/Turabian Style

Riccardo Fazio; Alessandra Jannelli. 2006. "Mathematical and numerical modeling for a bio-chemical aquarium." Applied Mathematics and Computation 174, no. 2: 1370-1383.