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Volume 13 (2022): Issue 1 (January 2022)

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Volume 10 (2019): Issue 2 (January 2019)
Special Issue on Mathematical Models and Methods in Biology, Medicine and Physiology. Guest Editors: Michele Piana, Luigi Preziosi

Volume 10 (2019): Issue 1 (February 2019)

Volume 9 (2018): Issue 2 (December 2018)
Special Issue on Mathematical modelling for complex systems: multi-agents methods. Guest Editor: Elena De Angelis

Volume 9 (2018): Issue 1 (February 2018)

Volume 8 (2017): Issue 1 (March 2017)

Volume 7 (2016): Issue 3 (September 2016)
"Special Issue on New Trends in Semi-Lagrangian Methods, Guest Editors: Luca Bonaventura, Maurizio Falcone and Roberto Ferretti

Volume 7 (2016): Issue 2 (June 2016)
Special Issue on Constitutive Equations for Heat Conduction in Nanosystems and Non-equilibrium Processes. Guest Editors: Vito Antonio Cimmelli and David Jou

Volume 7 (2016): Issue 1 (January 2016)
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Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

Volume 10 (2019): Issue 1 (February 2019)

Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

14 Articles
access type Open Access

Multi-Dimensional Chebyshev Polynomials: A Non-Conventional Approach

Published Online: 05 Feb 2019
Page range: 1 - 19

Abstract

Abstract

Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

Keywords

  • Translation operators
  • Hermite polynomials
  • Generating functions
  • Chebyshev polynomials
  • Gegenbauer polynomials
access type Open Access

Wigner Monte Carlo simulation without discretization error of the tunneling rectangular barrier

Published Online: 05 Feb 2019
Page range: 20 - 30

Abstract

Abstract

The Wigner transport equation can be solved stochastically by Monte Carlo techniques based on the theory of piecewise deterministic Markov processes. A new stochastic algorithm, without time discretization error, has been implemented and studied in the case of the quantum transport through a rectangular potential barrier.

Keywords

  • Nanostructures
  • Wigner transport equation
  • Direct simulation Monte Carlo
access type Open Access

Generalized special functions in the description of fractional diffusive equations

Published Online: 05 Feb 2019
Page range: 31 - 40

Abstract

Abstract

Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

Keywords

  • Hermite polynomials
  • heat equation
  • fractional calculus
access type Open Access

Improved mobility models for charge transport in graphene

Published Online: 11 May 2019
Page range: 41 - 52

Abstract

Abstract

Charge transport in graphene is crucial for the design of a new generation of nanoscale electron devices. A reasonable model is represented by the semiclassical Boltzmann equations for electrons in the valence and conduction bands. As shown by Romano et al. (J. Comput. Phys., 2015), the discontinuous Galerkin methods are a viable way to tackle the problem of the numerical integration of these equations, even if efficient DSMC with a proper inclusion of the Pauli principle have been also devised. One of the advantages of the solutions obtained with deterministic approach is of course the absence of statistical noise. This fact is crucial for an accurate estimation of the low field mobility as proved by Majorana et al. (J. Math. Industry, 2016) in the case of a unipolar charge transport in a suspended graphene sheet under a constant electric field.

The mobility expressions are essential for the drift-diffusion equations which constitute the most adopted models for charge transport in CAD. Here the analysis by Majorana et al. (J. Math. Industry, 2016) is improved in two ways: by including the charge transport both in the valence and conduction bands; by taking into account the presence of an oxide as substrate for the graphene sheet. New models of mobility are obtained and, in particular, relevant improvements of the low field mobility are achieved.

Keywords

  • Graphene
  • charge transport
  • mobility model
  • discontinuous Galerkin methods
access type Open Access

New Distance Concept and Graph Theory Approach for Certain Coding Techniques Design and Analysis

Published Online: 11 May 2019
Page range: 53 - 70

Abstract

Abstract

A New graph distance concept introduced for certain coding techniques helped in their design and analysis as in the case of distance-preserving mappings and spectral shaping codes. A graph theoretic construction, mapping binary sequences to permutation sequences and inspired from the k-cube graph has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. The new introduced distance concept in the k-cube graph helped better understanding and analyzing for the first time the concept of distance-reducing mappings. A combination of distance and the index-permutation graph concepts helped uncover and verify certain properties of spectral null codes, which were previously difficult to analyze.

Keywords

  • Graph theory
  • Distance mappings
  • Spectral codes
  • Cube graph
  • index permutation graph
access type Open Access

A degenerate pseudo-parabolic equation with memory

Published Online: 11 May 2019
Page range: 71 - 77

Abstract

Abstract

We prove the existence and uniqueness for a degenerate pseudo-parabolic problem with memory. This kind of problem arises in the study of the homogenization of some differential systems involving the Laplace-Beltrami operator and describes the effective behaviour of the electrical conduction in some composite materials.

Keywords

  • Existence and uniqueness
  • Pseudo-parabolic equations
  • Equations with memory
access type Open Access

On weak regularity requirements of the relaxation modulus in viscoelasticity

Published Online: 11 May 2019
Page range: 78 - 87

Abstract

Abstract

The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represented by a wedge continuous function. Indeed, the linear integro-differential viscoelasticity equation, characterised by a suitable wedge continuous relaxation function, is shown to give the classical linear wave equation via a limit procedure.

Keywords

  • Materials with memory
  • Viscoelasticity
  • hyperbolic integro-differential problem
  • Regular kernel integro-differential systems
  • Singular kernel integro-differential systems
access type Open Access

Mathematical Approach and Implementation of Frequency Mapping Techniques in Power-Line Communications Channel

Published Online: 11 May 2019
Page range: 88 - 108

Abstract

Abstract

Power-line channel is considered to be a very hostile channel compared to other channels in view of the different types of noise that could exist. Therefore, the choice of the error correcting code and the modulation scheme can play a big role in combating the noise in such a channel. M -FSK modulation has shown its robustness for such a type of channel. Two frequency mappings techniques are presented in this paper. In the first technique, M orthogonal frequencies are arranged in sequences based on the value and the position of permutation symbols, while in the second technique, the frequencies are rearranged based on the sign changes of the Walsh-Hadamard transform (WHT). The obtained M-FSK modulation is combined to codes based on Viterbi decoding algorithms since Viterbi decoder is considered to be the maximum-likelihood decoding algorithm for convolutional codes and codes with state machine representation. A mathematical approach and implementation of frequency mappings is introduced to investigate the performance of the new designed communication system in the presence of permanent frequency disturbances, also known as narrow-band interference (NBI), such as those encountered in power line communications (PLC) channel.

Keywords

  • -FSK Modulation
  • Power-Line Communications
  • Viterbi Decoder
  • Walsh-Hadamard transform
  • Mapping
access type Open Access

Quantum graphs and dimensional crossover: the honeycomb

Published Online: 15 Jun 2019
Page range: 109 - 122

Abstract

Abstract

We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schrödinger Equation on doubly-periodic metric graphs. We extend the results known for the two–dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one–dimensional and two–dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover.

Keywords

  • periodic graphs
  • nonlinear Schrödinger
  • threshold phenomena
  • Sobolev inequality
access type Open Access

A certified RB method for PDE-constrained parametric optimization problems

Published Online: 15 Jun 2019
Page range: 123 - 152

Abstract

Abstract

We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.

Keywords

  • PDE-constrained optimization
  • Reduced basis method
  • A posteriori error estimation
  • Adjoint-based approach
  • Non-convex cost functionals
access type Open Access

A BGK model for charge transport in graphene

Published Online: 12 Aug 2019
Page range: 153 - 161

Abstract

Abstract

The classical Boltzmann equation describes well temporal behaviour of a rarefied perfect gas. Modified kinetic equations have been proposed for studying the dynamics of different type of gases. An important example is the transport equation, which describes the charged particles flow, in the semi-classical regime, in electronic devices. In order to reduce the difficulties in solving the Boltzmann equation, simple expressions of a collision operator have been proposed to replace the standard Boltzmann integral term. These new equations are called kinetic models. The most popular and widely used kinetic model is the Bhatnagar-Gross-Krook (BGK) model. In this work we propose and analyse a BGK model for charge transport in graphene.

Keywords

  • Boltzmann equation
  • BGK model
  • graphene
access type Open Access

A study case for the analysis of asymptotic expansions beyond the tQSSA for inhibitory mechanisms in enzyme kinetics.

Published Online: 12 Aug 2019
Page range: 162 - 181

Abstract

Abstract

In this paper we study the model of the chemical reaction of fully competitive inhibition and determine the appropriate parameter (related to the chemical constants of the model), for the application of singular perturbation techniques. We determine the inner and the outer solutions up to the first perturbation order and the uniform expansions. Some numerical results are discussed.

Keywords

  • Enzyme Kinetics
  • Inhibition
  • Singular Perturbation Theory
  • Asymptotic Expansions
access type Open Access

Application of Energetic BEM to 2D Elastodynamic Soft Scattering Problems

Published Online: 18 Nov 2019
Page range: 182 - 198

Abstract

Abstract

Starting from a recently developed energetic space-time weak formulation of the Boundary Integral Equations related to scalar wave propagation problems, in this paper we focus for the first time on the 2D elastodynamic extension of the above wave propagation analysis. In particular, we consider elastodynamic scattering problems by open arcs, with vanishing initial and Dirichlet boundary conditions and we assess the efficiency and accuracy of the proposed method, on the basis of numerical results obtained for benchmark problems having available analytical solution.

Keywords

  • Elastodynamic wave propagation
  • soft scattering
  • space-time boundary integral equation
  • energetic boundary element method
access type Open Access

Effect of parameter selection on different topological structures for Particle Swarm Optimization algorithm

Published Online: 18 Nov 2019
Page range: 199 - 207

Abstract

Abstract

Particle Swarm Optimization is an evolutionary optimization algorithm, largely studied during the years: analysis of convergence, determination of the optimal coefficients, hybridization of the original algorithm and also the determination of the best relationship structure between the swarm elements (topology) have been investigated largely. Unfortunately, all these studies have been produced separately, and the same coefficients, derived for the original topology of the algorithm, have been always applied. The intent of this paper is to identify the best set of coefficients for different topological structures. A large suite of objective functions are considered and the best compromise coefficients are identified for each topology. Results are finally compared on the base of a practical ship design application.

Keywords

  • Optimization
  • Heuristic methods
  • Evolutionary computation
  • Particle Swarm Optimization
14 Articles
access type Open Access

Multi-Dimensional Chebyshev Polynomials: A Non-Conventional Approach

Published Online: 05 Feb 2019
Page range: 1 - 19

Abstract

Abstract

Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

Keywords

  • Translation operators
  • Hermite polynomials
  • Generating functions
  • Chebyshev polynomials
  • Gegenbauer polynomials
access type Open Access

Wigner Monte Carlo simulation without discretization error of the tunneling rectangular barrier

Published Online: 05 Feb 2019
Page range: 20 - 30

Abstract

Abstract

The Wigner transport equation can be solved stochastically by Monte Carlo techniques based on the theory of piecewise deterministic Markov processes. A new stochastic algorithm, without time discretization error, has been implemented and studied in the case of the quantum transport through a rectangular potential barrier.

Keywords

  • Nanostructures
  • Wigner transport equation
  • Direct simulation Monte Carlo
access type Open Access

Generalized special functions in the description of fractional diffusive equations

Published Online: 05 Feb 2019
Page range: 31 - 40

Abstract

Abstract

Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

Keywords

  • Hermite polynomials
  • heat equation
  • fractional calculus
access type Open Access

Improved mobility models for charge transport in graphene

Published Online: 11 May 2019
Page range: 41 - 52

Abstract

Abstract

Charge transport in graphene is crucial for the design of a new generation of nanoscale electron devices. A reasonable model is represented by the semiclassical Boltzmann equations for electrons in the valence and conduction bands. As shown by Romano et al. (J. Comput. Phys., 2015), the discontinuous Galerkin methods are a viable way to tackle the problem of the numerical integration of these equations, even if efficient DSMC with a proper inclusion of the Pauli principle have been also devised. One of the advantages of the solutions obtained with deterministic approach is of course the absence of statistical noise. This fact is crucial for an accurate estimation of the low field mobility as proved by Majorana et al. (J. Math. Industry, 2016) in the case of a unipolar charge transport in a suspended graphene sheet under a constant electric field.

The mobility expressions are essential for the drift-diffusion equations which constitute the most adopted models for charge transport in CAD. Here the analysis by Majorana et al. (J. Math. Industry, 2016) is improved in two ways: by including the charge transport both in the valence and conduction bands; by taking into account the presence of an oxide as substrate for the graphene sheet. New models of mobility are obtained and, in particular, relevant improvements of the low field mobility are achieved.

Keywords

  • Graphene
  • charge transport
  • mobility model
  • discontinuous Galerkin methods
access type Open Access

New Distance Concept and Graph Theory Approach for Certain Coding Techniques Design and Analysis

Published Online: 11 May 2019
Page range: 53 - 70

Abstract

Abstract

A New graph distance concept introduced for certain coding techniques helped in their design and analysis as in the case of distance-preserving mappings and spectral shaping codes. A graph theoretic construction, mapping binary sequences to permutation sequences and inspired from the k-cube graph has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. The new introduced distance concept in the k-cube graph helped better understanding and analyzing for the first time the concept of distance-reducing mappings. A combination of distance and the index-permutation graph concepts helped uncover and verify certain properties of spectral null codes, which were previously difficult to analyze.

Keywords

  • Graph theory
  • Distance mappings
  • Spectral codes
  • Cube graph
  • index permutation graph
access type Open Access

A degenerate pseudo-parabolic equation with memory

Published Online: 11 May 2019
Page range: 71 - 77

Abstract

Abstract

We prove the existence and uniqueness for a degenerate pseudo-parabolic problem with memory. This kind of problem arises in the study of the homogenization of some differential systems involving the Laplace-Beltrami operator and describes the effective behaviour of the electrical conduction in some composite materials.

Keywords

  • Existence and uniqueness
  • Pseudo-parabolic equations
  • Equations with memory
access type Open Access

On weak regularity requirements of the relaxation modulus in viscoelasticity

Published Online: 11 May 2019
Page range: 78 - 87

Abstract

Abstract

The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represented by a wedge continuous function. Indeed, the linear integro-differential viscoelasticity equation, characterised by a suitable wedge continuous relaxation function, is shown to give the classical linear wave equation via a limit procedure.

Keywords

  • Materials with memory
  • Viscoelasticity
  • hyperbolic integro-differential problem
  • Regular kernel integro-differential systems
  • Singular kernel integro-differential systems
access type Open Access

Mathematical Approach and Implementation of Frequency Mapping Techniques in Power-Line Communications Channel

Published Online: 11 May 2019
Page range: 88 - 108

Abstract

Abstract

Power-line channel is considered to be a very hostile channel compared to other channels in view of the different types of noise that could exist. Therefore, the choice of the error correcting code and the modulation scheme can play a big role in combating the noise in such a channel. M -FSK modulation has shown its robustness for such a type of channel. Two frequency mappings techniques are presented in this paper. In the first technique, M orthogonal frequencies are arranged in sequences based on the value and the position of permutation symbols, while in the second technique, the frequencies are rearranged based on the sign changes of the Walsh-Hadamard transform (WHT). The obtained M-FSK modulation is combined to codes based on Viterbi decoding algorithms since Viterbi decoder is considered to be the maximum-likelihood decoding algorithm for convolutional codes and codes with state machine representation. A mathematical approach and implementation of frequency mappings is introduced to investigate the performance of the new designed communication system in the presence of permanent frequency disturbances, also known as narrow-band interference (NBI), such as those encountered in power line communications (PLC) channel.

Keywords

  • -FSK Modulation
  • Power-Line Communications
  • Viterbi Decoder
  • Walsh-Hadamard transform
  • Mapping
access type Open Access

Quantum graphs and dimensional crossover: the honeycomb

Published Online: 15 Jun 2019
Page range: 109 - 122

Abstract

Abstract

We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schrödinger Equation on doubly-periodic metric graphs. We extend the results known for the two–dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one–dimensional and two–dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover.

Keywords

  • periodic graphs
  • nonlinear Schrödinger
  • threshold phenomena
  • Sobolev inequality
access type Open Access

A certified RB method for PDE-constrained parametric optimization problems

Published Online: 15 Jun 2019
Page range: 123 - 152

Abstract

Abstract

We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.

Keywords

  • PDE-constrained optimization
  • Reduced basis method
  • A posteriori error estimation
  • Adjoint-based approach
  • Non-convex cost functionals
access type Open Access

A BGK model for charge transport in graphene

Published Online: 12 Aug 2019
Page range: 153 - 161

Abstract

Abstract

The classical Boltzmann equation describes well temporal behaviour of a rarefied perfect gas. Modified kinetic equations have been proposed for studying the dynamics of different type of gases. An important example is the transport equation, which describes the charged particles flow, in the semi-classical regime, in electronic devices. In order to reduce the difficulties in solving the Boltzmann equation, simple expressions of a collision operator have been proposed to replace the standard Boltzmann integral term. These new equations are called kinetic models. The most popular and widely used kinetic model is the Bhatnagar-Gross-Krook (BGK) model. In this work we propose and analyse a BGK model for charge transport in graphene.

Keywords

  • Boltzmann equation
  • BGK model
  • graphene
access type Open Access

A study case for the analysis of asymptotic expansions beyond the tQSSA for inhibitory mechanisms in enzyme kinetics.

Published Online: 12 Aug 2019
Page range: 162 - 181

Abstract

Abstract

In this paper we study the model of the chemical reaction of fully competitive inhibition and determine the appropriate parameter (related to the chemical constants of the model), for the application of singular perturbation techniques. We determine the inner and the outer solutions up to the first perturbation order and the uniform expansions. Some numerical results are discussed.

Keywords

  • Enzyme Kinetics
  • Inhibition
  • Singular Perturbation Theory
  • Asymptotic Expansions
access type Open Access

Application of Energetic BEM to 2D Elastodynamic Soft Scattering Problems

Published Online: 18 Nov 2019
Page range: 182 - 198

Abstract

Abstract

Starting from a recently developed energetic space-time weak formulation of the Boundary Integral Equations related to scalar wave propagation problems, in this paper we focus for the first time on the 2D elastodynamic extension of the above wave propagation analysis. In particular, we consider elastodynamic scattering problems by open arcs, with vanishing initial and Dirichlet boundary conditions and we assess the efficiency and accuracy of the proposed method, on the basis of numerical results obtained for benchmark problems having available analytical solution.

Keywords

  • Elastodynamic wave propagation
  • soft scattering
  • space-time boundary integral equation
  • energetic boundary element method
access type Open Access

Effect of parameter selection on different topological structures for Particle Swarm Optimization algorithm

Published Online: 18 Nov 2019
Page range: 199 - 207

Abstract

Abstract

Particle Swarm Optimization is an evolutionary optimization algorithm, largely studied during the years: analysis of convergence, determination of the optimal coefficients, hybridization of the original algorithm and also the determination of the best relationship structure between the swarm elements (topology) have been investigated largely. Unfortunately, all these studies have been produced separately, and the same coefficients, derived for the original topology of the algorithm, have been always applied. The intent of this paper is to identify the best set of coefficients for different topological structures. A large suite of objective functions are considered and the best compromise coefficients are identified for each topology. Results are finally compared on the base of a practical ship design application.

Keywords

  • Optimization
  • Heuristic methods
  • Evolutionary computation
  • Particle Swarm Optimization

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