Axioms

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4 pages, 181 KiB

Open AccessEditorial

Differential Models, Numerical Simulations and Applications

by Gabriella Bretti

Axioms 2021, 10(4), 260; https://doi.org/10.3390/axioms10040260 - 19 Oct 2021

Cited by 2 | Viewed by 1476

Abstract

Differential models, numerical methods and computer simulations play a fundamental role in applied sciences. Since most of the differential models inspired by real world applications have no analytical solutions, the development of numerical methods and efficient simulation algorithms play a key role in [...] Read more.

Differential models, numerical methods and computer simulations play a fundamental role in applied sciences. Since most of the differential models inspired by real world applications have no analytical solutions, the development of numerical methods and efficient simulation algorithms play a key role in the computation of the solutions to many relevant problems. Moreover, since the model parameters in mathematical models have interesting scientific interpretations and their values are often unknown, estimation techniques need to be developed for parameter identification against the measured data of observed phenomena. In this respect, this Special Issue collects some important developments in different areas of application. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

Research

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30 pages, 1720 KiB

Open AccessArticle

Estimation Algorithm for a Hybrid PDE–ODE Model Inspired by Immunocompetent Cancer-on-Chip Experiment

by Gabriella Bretti, Adele De Ninno, Roberto Natalini, Daniele Peri and Nicole Roselli

Axioms 2021, 10(4), 243; https://doi.org/10.3390/axioms10040243 - 28 Sep 2021

Cited by 13 | Viewed by 2894

Abstract

The present work is motivated by the development of a mathematical model mimicking the mechanisms observed in lab-on-chip experiments, made to reproduce on microfluidic chips the in vivo reality. Here we consider the Cancer-on-Chip experiment where tumor cells are treated with chemotherapy drug [...] Read more.

The present work is motivated by the development of a mathematical model mimicking the mechanisms observed in lab-on-chip experiments, made to reproduce on microfluidic chips the in vivo reality. Here we consider the Cancer-on-Chip experiment where tumor cells are treated with chemotherapy drug and secrete chemical signals in the environment attracting multiple immune cell species. The in silico model here proposed goes towards the construction of a “digital twin” of the experimental immune cells in the chip environment to better understand the complex mechanisms of immunosurveillance. To this aim, we develop a tumor-immune microfluidic hybrid PDE–ODE model to describe the concentration of chemicals in the Cancer-on-Chip environment and immune cells migration. The development of a trustable simulation algorithm, able to reproduce the immunocompetent dynamics observed in the chip, requires an efficient tool for the calibration of the model parameters. In this respect, the present paper represents a first methodological work to test the feasibility and the soundness of the calibration technique here proposed, based on a multidimensional spline interpolation technique for the time-varying velocity field surfaces obtained from cell trajectories. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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21 pages, 945 KiB

Open AccessFeature PaperArticle

Relaxation Limit of the Aggregation Equation with Pointy Potential

by Benoît Fabrèges, Frédéric Lagoutière, Sébastien Tran Tien and Nicolas Vauchelet

Axioms 2021, 10(2), 108; https://doi.org/10.3390/axioms10020108 - 28 May 2021

Cited by 2 | Viewed by 2688

Abstract

This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through [...] Read more.

This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigated an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigated the numerical discretization of this relaxation limit by uniformly accurate schemes. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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21 pages, 3223 KiB

Open AccessArticle

Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study

by Maya Briani, Emiliano Cristiani and Paolo Ranut

Axioms 2021, 10(2), 102; https://doi.org/10.3390/axioms10020102 - 24 May 2021

Cited by 2 | Viewed by 2914

Abstract

In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that [...] Read more.

In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions, the creeping phenomenon can appear, i.e., one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic follow-the-leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena such as stop and go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste–Venice and its branches, provided by Autovie Venete S.p.A. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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16 pages, 864 KiB

Open AccessArticle

An Information-Theoretic Framework for Optimal Design: Analysis of Protocols for Estimating Soft Tissue Parameters in Biaxial Experiments

by Ankush Aggarwal, Damiano Lombardi and Sanjay Pant

Axioms 2021, 10(2), 79; https://doi.org/10.3390/axioms10020079 - 1 May 2021

Cited by 5 | Viewed by 2672

Abstract

A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information and their combination is proposed. The framework is tested on the analysis of protocols—a combination of angles along which strain measurements can be acquired—in a biaxial [...] Read more.

A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information and their combination is proposed. The framework is tested on the analysis of protocols—a combination of angles along which strain measurements can be acquired—in a biaxial experiment of soft tissues for the estimation of hyperelastic constitutive model parameters. The proposed framework considers the information gain about the parameters from the experiment as the key criterion to be maximised, which can be directly used for optimal design. Information gain is computed through k-nearest neighbour algorithms applied to the joint samples of the parameters and measurements produced by the forward and observation models. For biaxial experiments, the results show that low angles have a relatively low information content compared to high angles. The results also show that a smaller number of angles with suitably chosen combinations can result in higher information gains when compared to a larger number of angles which are poorly combined. Finally, it is shown that the proposed framework is consistent with classical approaches, particularly D-optimal design. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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14 pages, 3120 KiB

Open AccessFeature PaperArticle

A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations

by Maria Francesca Carfora and Isabella Torcicollo

Axioms 2021, 10(2), 78; https://doi.org/10.3390/axioms10020078 - 30 Apr 2021

Cited by 5 | Viewed by 2760

Abstract

A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by [...] Read more.

A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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9 pages, 260 KiB

Open AccessArticle

A Quadratic Mean Field Games Model for the Langevin Equation

by Fabio Camilli

Axioms 2021, 10(2), 68; https://doi.org/10.3390/axioms10020068 - 19 Apr 2021

Cited by 3 | Viewed by 1801

Abstract

We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. An appropriate change of variables transforms the Mean Field Games system into a system of two coupled kinetic [...] Read more.

We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. An appropriate change of variables transforms the Mean Field Games system into a system of two coupled kinetic Fokker–Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

21 pages, 999 KiB

Open AccessFeature PaperArticle

Non-Standard Discrete RothC Models for Soil Carbon Dynamics

by Fasma Diele, Carmela Marangi and Angela Martiradonna

Axioms 2021, 10(2), 56; https://doi.org/10.3390/axioms10020056 - 8 Apr 2021

Cited by 6 | Viewed by 3166

Abstract

Soil Organic Carbon (SOC) is one of the key indicators of land degradation. SOC positively affects soil functions with regard to habitats, biological diversity and soil fertility; therefore, a reduction in the SOC stock of soil results in degradation, and it may also [...] Read more.

Soil Organic Carbon (SOC) is one of the key indicators of land degradation. SOC positively affects soil functions with regard to habitats, biological diversity and soil fertility; therefore, a reduction in the SOC stock of soil results in degradation, and it may also have potential negative effects on soil-derived ecosystem services. Dynamical models, such as the Rothamsted Carbon (RothC) model, may predict the long-term behaviour of soil carbon content and may suggest optimal land use patterns suitable for the achievement of land degradation neutrality as measured in terms of the SOC indicator. In this paper, we compared continuous and discrete versions of the RothC model, especially to achieve long-term solutions. The original discrete formulation of the RothC model was then compared with a novel non-standard integrator that represents an alternative to the exponential Rosenbrock–Euler approach in the literature. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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15 pages, 1402 KiB

Open AccessArticle

The Role of Spectral Complexity in Connectivity Estimation

by Elisabetta Vallarino, Alberto Sorrentino, Michele Piana and Sara Sommariva

Axioms 2021, 10(1), 35; https://doi.org/10.3390/axioms10010035 - 16 Mar 2021

Cited by 6 | Viewed by 2408

Abstract

The study of functional connectivity from magnetoecenphalographic (MEG) data consists of quantifying the statistical dependencies among time series describing the activity of different neural sources from the magnetic field recorded outside the scalp. This problem can be addressed by utilizing connectivity measures whose [...] Read more.

The study of functional connectivity from magnetoecenphalographic (MEG) data consists of quantifying the statistical dependencies among time series describing the activity of different neural sources from the magnetic field recorded outside the scalp. This problem can be addressed by utilizing connectivity measures whose computation in the frequency domain often relies on the evaluation of the cross-power spectrum of the neural time series estimated by solving the MEG inverse problem. Recent studies have focused on the optimal determination of the cross-power spectrum in the framework of regularization theory for ill-posed inverse problems, providing indications that, rather surprisingly, the regularization process that leads to the optimal estimate of the neural activity does not lead to the optimal estimate of the corresponding functional connectivity. Along these lines, the present paper utilizes synthetic time series simulating the neural activity recorded by an MEG device to show that the regularization of the cross-power spectrum is significantly correlated with the signal-to-noise ratio of the measurements and that, as a consequence, this regularization correspondingly depends on the spectral complexity of the neural activity. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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21 pages, 1718 KiB

Open AccessFeature PaperArticle

A Cellular Potts Model for Analyzing Cell Migration across Constraining Pillar Arrays

by Marco Scianna and Luigi Preziosi

Axioms 2021, 10(1), 32; https://doi.org/10.3390/axioms10010032 - 12 Mar 2021

Cited by 6 | Viewed by 3317

Abstract

Cell migration in highly constrained environments is fundamental in a wide variety of physiological and pathological phenomena. In particular, it has been experimentally shown that the migratory capacity of most cell lines depends on their ability to transmigrate through narrow constrictions, which in [...] Read more.

Cell migration in highly constrained environments is fundamental in a wide variety of physiological and pathological phenomena. In particular, it has been experimentally shown that the migratory capacity of most cell lines depends on their ability to transmigrate through narrow constrictions, which in turn relies on their deformation capacity. In this respect, the nucleus, which occupies a large fraction of the cell volume and is substantially stiffer than the surrounding cytoplasm, imposes a major obstacle. This aspect has also been investigated with the use of microfluidic devices formed by dozens of arrays of aligned polymeric pillars that limit the available space for cell movement. Such experimental systems, in particular, in the designs developed by the groups of Denais and of Davidson, were here reproduced with a tailored version of the Cellular Potts model, a grid-based stochastic approach where cell dynamics are established by a Metropolis algorithm for energy minimization. The proposed model allowed quantitatively analyzing selected cell migratory determinants (e.g., the cell and nuclear speed and deformation, and forces acting at the nuclear membrane) in the case of different experimental setups. Most of the numerical results show a remarkable agreement with the corresponding empirical data. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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11 pages, 825 KiB

Open AccessFeature PaperArticle

Analysis of the Transient Behaviour in the Numerical Solution of Volterra Integral Equations

by Eleonora Messina and Antonia Vecchio

Axioms 2021, 10(1), 23; https://doi.org/10.3390/axioms10010023 - 23 Feb 2021

Cited by 1 | Viewed by 1885

Abstract

In this paper, the asymptotic behaviour of the numerical solution to the Volterra integral equations is studied. In particular, a technique based on an appropriate splitting of the kernel is introduced, which allows one to obtain vanishing asymptotic (transient) behaviour in the numerical [...] Read more.

In this paper, the asymptotic behaviour of the numerical solution to the Volterra integral equations is studied. In particular, a technique based on an appropriate splitting of the kernel is introduced, which allows one to obtain vanishing asymptotic (transient) behaviour in the numerical solution, consistently with the properties of the analytical solution, without having to operate restrictions on the integration steplength. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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20 pages, 2865 KiB

Open AccessFeature PaperArticle

A Three-Phase Fundamental Diagram from Three-Dimensional Traffic Data

by Maria Laura Delle Monache, Karen Chi, Yong Chen, Paola Goatin, Ke Han, Jing-mei Qiu and Benedetto Piccoli

Axioms 2021, 10(1), 17; https://doi.org/10.3390/axioms10010017 - 7 Feb 2021

Cited by 5 | Viewed by 4566

Abstract

This paper uses empirical traffic data collected from three locations in Europe and the US to reveal a three-phase fundamental diagram with two phases located in the uncongested regime. Model-based clustering, hypothesis testing and regression analyses are applied to the speed–flow–occupancy relationship represented [...] Read more.

This paper uses empirical traffic data collected from three locations in Europe and the US to reveal a three-phase fundamental diagram with two phases located in the uncongested regime. Model-based clustering, hypothesis testing and regression analyses are applied to the speed–flow–occupancy relationship represented in the three-dimensional space to rigorously validate the three phases and identify their gaps. The finding is consistent across the aforementioned different geographical locations. Accordingly, we propose a three-phase macroscopic traffic flow model and a characterization of solutions to the Riemann problems. This work identifies critical structures in the fundamental diagram that are typically ignored in first- and higher-order models and could significantly impact travel time estimation on highways. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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20 pages, 1044 KiB

Open AccessArticle

Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity

by Simone Göttlich, Michael Herty and Gediyon Weldegiyorgis

Axioms 2021, 10(1), 12; https://doi.org/10.3390/axioms10010012 - 21 Jan 2021

Cited by 2 | Viewed by 3054

Abstract

In this paper, we study input-to-state stability (ISS) of an equilibrium for a scalar conservation law with nonlocal velocity and measurement error arising in a highly re-entrant manufacturing system. By using a suitable Lyapunov function, we prove sufficient and necessary conditions on ISS. [...] Read more.

In this paper, we study input-to-state stability (ISS) of an equilibrium for a scalar conservation law with nonlocal velocity and measurement error arising in a highly re-entrant manufacturing system. By using a suitable Lyapunov function, we prove sufficient and necessary conditions on ISS. We propose a numerical discretization of the scalar conservation law with nonlocal velocity and measurement error. A suitable discrete Lyapunov function is analyzed to provide ISS of a discrete equilibrium for the proposed numerical approximation. Finally, we show computational results to validate the theoretical findings. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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