Showing papers in "Mathematical Modelling of Natural Phenomena in 2019"

New aspects of fractional Biswas–Milovic model with Mittag-Leffler law

Abstract: This article deals with a fractional extension of Biswas–Milovic (BM) model having Kerr and parabolic law nonlinearities. The BM model plays a key role in describing the long-distance optical communications. The fractional homotopy analysis transform technique (FHATM) is applied to examine the BM equation involving Atangana–Baleanu (AB) derivative of fractional order. The FHATM is constructed by using homotopy analysis technique, Laplace transform algorithm and homotopy polynomials. The numerical simulation work is performed with the aid of maple software package. In order to demonstrate the effects of order of AB operator, variables and parameters on the displacement, the results are shown graphically. The outcomes of the present investigation are very encouraging and show that the AB fractional operator is very useful in mathematical modelling of natural phenomena.

Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative

Abstract: A virus that causes hepatitis E is known as (HEV) and regarded on of the reason for lever inflammation. In mathematical aspects a very low attention has been paid to HEV dynamics. Therefore, the present work explores the HEV dynamics in fractional derivative. The Caputo–Fabriizo derivative is used to study the dynamics of HEV. First, the essential properties of the model will be presented and then describe the HEV model with CF derivative. Application of fixed point theory is used to obtain the existence and uniqueness results associated to the model. By using Adams–Bashfirth numerical scheme the solution is obtained. Some numerical results and tables for arbitrary order derivative are presented.

A hybrid analytical algorithm for nonlinear fractional wave-like equations

Abstract: In this work, we aim to present a hybrid numerical scheme based on the homotopy analysis transform method (HATM) to examine the fractional model of nonlinear wave-like equations having variable coefficients, which narrate the evolution of stochastic systems. The wave-like equation models the erratic motions of small particles that are dipped in fluids and fluctuations of the stochastic behavior of exchange rates. The uniqueness and existence of HATM solution have also been discussed. Some numerical examples are given to establish the accurateness and effectiveness of the suggested scheme. Furthermore, we show that the proposed computational approach can give much better approximation than perturbation and Adomain decomposition method, which are the special cases of HATM. The result exhibits that the HATM is very productive, straight out and computationally very attractive.

Characterizations of two different fractional operators without singular kernel

Abstract: In this paper, we analyze the behaviours of two different fractional derivative operators defined in the last decade. One of them is defined with the normalized sinc function (NSF) and the other one is defined with the Mittag-Leffler function (MLF). Both of them have a non-singular kernel. The fractional derivative operator defined with the MLF is developed by Atangana and Baleanu (ABO) in 2016 and the other operator defined with the normalized sinc function (NSFDO) is created by Yang et al. in 2017. These mentioned operators have some advantages to model the real life problems and to solve them. On the other hand, since the Laplace transform (LT) of the ABO can be calculated more easily, it can be preferred to solve linear/nonlinear problems. In this study, we use the perturbation method with coupled the LTs of these operators to analyze their performance in solving some fractional differential equations. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

Fractional order model of immune cells influenced by cancer cells

Abstract: In this paper, we study the mathematical model of interaction cancer cells and immune system cells presented Castiglione and Piccoli. As the interaction between cancer cells and the immune system is weak, when the immune system of the body begins to decrease, the cancer cells get stronger and increase rapidly. Helper CD4+ T and cytotoxic CD8+ T cells, cancer cells, dendritic cells and cytokine interleukin-2 (IL-2) cells are involved in the mathematical model of this competition in the living body. As can be seen in the literature, since the cancer cells have memory structure, fractional models describe the struggle between the cancer cells and immune system give more meaningful results than classical models as closer to the reality. The main motivation of the present work is to generalize the model in Castiglione and Piccoli [J. Theor. Biol. 247 (2007) 723–732] by using Caputo fractional derivative. The main aim is to analyze the behaviors of system cells by changing of the fractional parameter. In this sense, we study on the stability analysis of treatment free and the fixed points of the prescribed model. To get the numerical solutions, we apply the Adam-Bashforth-Moulton (ABM) algorithm and also illustrate the results by the graphics held by Matlab program. We have reached the excellent result that cancer cells decrease as θ diminishes in this process.

Response functions in linear viscoelastic constitutive equations and related fractional operators

Abstract: This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

Modeling and analysis of fractional neutral disturbance waves in arterial vessels

Abstract: The behavior of neutral disturbance in arterial vessels has attracted more and more attention in recent decades because it carries some important information which can be applied to predict and diagnose related heart disease, such as arteriosclerosis and hypertension, etc. Because of the complexity of blood flow in arteries, it is very necessary to construct accurate mathematical model and analyze the mechanical behavior of neutral disturbance in arterial vessels. In this paper, start from the basic equations of blood flow and the two-dimensional Navier–Stokes equation, the vorticity equation describing the disturbance flow is presented. Then, by use of multi-scale analysis and perturbation expansion method, the ZK equation is put forward which can reflect the behavior of the neutral perturbation flow in arterial vessels. Compared with the traditional KdV model, the model established in the paper can show the propagation of the disturbance flow in the radius direction. Furthermore, the time-fractional ZK equation is derived by semi-inverse method and Agrawal’s method, which is more convenient and accurate for discussing the feature of neutral disturbance in arterial vessels and can provide more information for analyzing some related heart disease. Meanwhile, with the help of the modified extended tanh method, the above mentioned equation is solved. The results show that neutral disturbance exists in arterial vessels and propagates in the form of solitary waves. By calculating, we find the relation of the stroke volume with vascular radius, blood flow velocity as well as the fractional order parameter α , which is very meaningful for preventing and treating related heart disease because the stroke volume is closely linked with heart disease.

Mathematical Modelling of Atherosclerosis

Abstract: The review presents the state of the art in the atherosclerosis modelling. It begins with the biological introduction describing the mechanisms of chronic inflammation of artery walls characterizing the development of atherosclerosis. In particular, we present in more detail models describing this chronic inflammation as a reaction-diffusion wave with regimes of propagation depending on the level of cholesterol (LDL) and models of rolling monocytes initializing the inflammation. Further development of this disease results in the formation of atherosclerotic plaque, vessel remodelling and possible plaque rupture due its interaction with blood flow. We review plaque-flow interaction models as well as reduced models (0D and 1D) of blood flow in atherosclerotic vasculature.

Immune response in HIV epidemics for distinct transmission rates and for saturated CTL response

Abstract: In this paper, we study the immune response in a fractional order model for HIV dynamics, for distinct disease transmission rates and saturated cytotoxic T-lymphocyte (CTL) response. Our goal is twofold: (i) to analyze the role of the order of the fractional derivative, α , on the efficacy of the immune response, (ii) to examine the immune response for distinct transmission functions, in the presence of saturated CTL response. We compute the reproduction number of the model and state the stability of the disease-free equilibrium. We discuss the results of the model from an epidemiological point of view.

Initial-boundary value problems for a time-fractional differential equation with involution perturbation

Abstract: Direct and inverse initial-boundary value problems of a time-fractional heat equation with involution perturbation are considered using both local and nonlocal boundary conditions. Results on existence of formal solutions to these problems are presented. Solutions are expressed in a form of series expansions using appropriate orthogonal basis obtained by separation of variables. Convergence of series solutions are obtained by imposing certain conditions on the given data. Uniqueness of the obtained solutions are also discussed. The obtained general solutions are illustrated by an example using an appropriate choice of the given data.

Nonlocal problems and functional-differential equations: theoretical aspects and applications to mathematical modelling

Abstract: This paper presents a review of results on nonlocal problems, functional-differential equations, and their applications, obtained during several last years. The following research areas are covered: the Kato square root problem for functional-differential operators, Vlasov equations and their applications to the modelling of high-temperature plasma, specific properties of differential-difference equations with incommensurable translations, degenerate functional-differential equations and their applications, functional-differential equations with contractions and extensions of independent variables, and operational methods for parabolic and elliptic functional-differential equations.

Distributed optimal control models in environmental economics: a review

Abstract: We review the most recent advances in distributed optimal control applied to Environmental Economics, covering in particular problems where the state dynamics are governed by partial differential equations (PDEs). This is a quite fresh application area of distributed optimal control, which has already suggested several new mathematical research lines due to the specificities of the Environmental Economics problems involved. We enhance the latter through a survey of the variety of themes and associated mathematical structures beared by this literature. We also provide a quick tour of the existing tools in the theory of distributed optimal control that have been applied so far in Environmental Economics.

The existence and asymptotic stability of periodic solutions with an interior layer of Burgers type equations with modular advection

Abstract: We consider a new class of singularly perturbed parabolic periodic boundary value problems for reaction-advection-diffusion equations: Burgers type equations with modular advection. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. The asymptotic stability of this solution is also established.

Mathematical modeling of frontal polymerization

Abstract: Frontal polymerization is way to convert liquid resin into a solid material with a self-propagating reaction. The reaction spreads like a flame from the heat of the reaction that diffuses into neighboring regions, starting more reaction. The frontal velocity has been accurately modeled for free-radical polymerization systems. The dynamics of fronts have been studied theoretically and experimentally. If the viscosity of the initial medium is low, then fronts can become unstable due to buoyancy-driven convection. A fascinating aspect of frontal polymerization is that fronts often do not propagate as a plane waves but exhibit complex modes such as “spin modes” and chaos. The kinetics of the polymerization significantly affects the onset of these modes. Multifunctional acrylates exhibit more complex dynamics than monoacrylates. Using multifunctional acrylates and inorganic fillers, 3P LLC created “cure-on demand” systems that do not require mixing before use, have a long shelf life and can be hardened in seconds to minutes. We consider two commercial products using frontal polymerization. The first is a wood filler that can be applied to a damaged section of wood and hardened in a few seconds by the application of heat to the surface. The second product is QuickCure Clay (QCC). QCC has an unlimited working time during which it can be sculpted. QCC is then cured by heating part of the object to 100 ° C, setting off the propagating curing front. The modeling of frontal polymerization helped guide the development of these products.

A new approach to capture heterogeneity in groundwater problem: An illustration with an Earth equation

Abstract: One of the major problem faced in modeling groundwater flow problems is perhaps how to capture heterogeneity of the geological formation within which the flow takes place. In this paper, we suggested applied a newly established approach to model real world problems that combines the concept of stochastic modeling in which parameters inputs are converted into distributions and the time differential operator is replaced by non-local differential operators. We illustrated this method with the Earth equation of groundwater recharge. For each case, we provided numerical and exact solution using the newly established numerical scheme and Laplace transform. We presented some numerical simulations. The numerical graphical representations let no doubt to think that this approach is the future way of modeling complex problems.

Multiscale dynamics of an adaptive catalytic network

Abstract: We study the multiscale structure of the Jain–Krishna adaptive network model. This model describes the co-evolution of a set of continuous-time autocatalytic ordinary differential equations and its underlying discrete-time graph structure. The graph dynamics is governed by deletion of vertices with asymptotically weak concentrations of prevalence and then re-insertion of vertices with new random connections. In this work, we prove several results about convergence of the continuous-time dynamics to equilibrium points. Furthermore, we motivate via formal asymptotic calculations several conjectures regarding the discrete-time graph updates. In summary, our results clearly show that there are several time scales in the problem depending upon system parameters, and that analysis can be carried out in certain singular limits. This shows that for the Jain–Krishna model, and potentially many other adaptive network models, a mixture of deterministic and/or stochastic multiscale methods is a good approach to work towards a rigorous mathematical analysis.

An infection age-space-structured SIR epidemic model with Dirichlet boundary condition

Abstract: In this paper, we are concerned with the global asymptotic behavior of an SIR epidemic model with infection age-space structure. Under the homogeneous Dirichlet boundary condition, we first reformulate the model into the coupled reaction-diffusion and difference system by using the method of characteristics. We then obtain the spatially heterogeneous disease-free steady state and define the basic reproduction number ℛ 0 by the spectral radius of the next generation operator. We then show the existence and uniqueness of the global classical solution by constructing suitable upper and lower solutions. As a threshold result, we establish that the disease-free steady state is globally attractive if ℛ 0 0 > 1. Finally, numerical simulations are exhibited to illustrate our theoretical results together with how to compute ℛ 0 .

Stabilization of a viscoelastic Timoshenko beam fixed into a moving base

Abstract: In this paper, we are concerned with a cantilevered Timoshenko beam. The beam is viscoelastic and subject to a translational displacement. Consequently, the Timoshenko system is complemented by an ordinary differential equation describing the dynamic of the base to which the beam is attached to. We establish a control force capable of driving the system to the equilibrium state with a certain speed depending on the decay rate of the relaxation function.

Geographic environmental Kuznets curves: the optimal growth linear-quadratic case

Abstract: We solve a linear-quadratic model of a spatio-temporal economy using a polluting one-input technology. Space is continuous and heterogenous: locations differ in productivity, nature self-cleaning technology and environmental awareness. The unique link between locations is transboundary pollution which is modelled as a PDE diffusion equation. The spatio-temporal functional is quadratic in local consumption and linear in pollution. Using a dynamic programming method adapted to our infinite dimensional setting, we solve the associated optimal control problem in closed-form and identify the asymptotic (optimal) spatial distribution of pollution. We show that optimal emissions will decrease at given location if and only if local productivity is larger than a threshold which depends both on the local pollution absorption capacity and environmental awareness. Furthermore, we numerically explore the relationship between the spatial optimal distributions of production and (asymptotic) pollution in order to uncover possible (geographic) environmental Kuznets curve cases.

Propagation of two independent sources of uncertainty in the electrocardiography imaging inverse solution

Abstract: This study investigates the effects of the input parameter uncertainties (organ conductivities, boundary data, etc.) on the electrocardiography (ECG) imaging problem. These inputs are very important for the construction of the torso potential for the forward problem and for the non-invasive electrical potential on the heart surface in the case of the inverse problem.We propose a new stochastic formulation that allows us to combine both sources of errors. We formulate the forward and inverse stochastic problems by considering the input parameters as random fields and a stochastic optimal control formulation. In order to quantify multiple independent sources of uncertainties on the forward and inverse solutions, we attribute suitable probability density functions for each randomness source and apply stochastic finite elements based on generalized polynomial chaos method. The efficiency of this approach to solve the forward and inverse ECG problems and the usability to quantify the effect of organ conductivity and epicardial boundary data uncertainties in the torso are demonstrated by a number of numerical simulations on a two-dimensional computational mesh of a realistic torso geometry.

A nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology

Abstract: We consider the identification problem of the conductivity coefficient for an elliptic operator using an incomplete over specified measures on the surface. Our purpose is to introduce an original method based on a game theory approach, and design a new algorithm for the simultaneous identification of conductivity coefficient and data completion process. We define three players with three corresponding criteria. The two first players use Dirichlet and Neumann strategies to solve the completion problem, while the third one uses the conductivity coefficient as strategy, and uses a cost which basically relies on an identifiability theorem. In our work, the numerical experiments seek the development of this algorithm for the electrocardiography imaging inverse problem, dealing with in-homogeneities in the torso domain. Furthermore, in our approach, the conductivity coefficients are known only by an approximate values. we conduct numerical experiments on a 2D torso case including noisy measurements. Results illustrate the ability of our computational approach to tackle the difficult problem of joint identification and data completion. Mathematics Subject Classification. 35J25, 35N05, 91A80. The dates will be set by the publisher.

The optimal population size under pollution and migration externalities: A spatial control approach

Abstract: We analyze the implications of pollution and migration externalities on the optimal population dynamics in a spatial setting. We focus on a framework in which pollution affects the mortality rate and decreases utility, while migration occurs within the spatial economy. Agents optimally determine their fertility rate which, along with pollution-induced mortality and spatial migration, determines the net population growth rate. This setting implies that human population follows an endogenous logistic-type dynamics where fertility choices determine what the optimal limit of human population will be. We compare the decentralized and the centralized outcomes showing that such fertility decisions generally differ, quantifying the extent to which pollution and migration induced externalities matter in determining the difference between the two outcomes. We show that, due to the effects of pollution on utility and mortality, both the optimal fertility rate and the population size are smallest in the centralized economy but migration effects change not only the size of these differences but also their direction, suggesting that the spatial channel is an important mechanism to account for in the process of policymaking.

Black swans and canards in two predator – one prey model

Abstract: In this paper, we show how canards can be easily caught in a class of 3D systems with an exact black swan (a slow invariant manifold of variable stability). We demonstrate this approach to a canard chase via the two predator – one prey model. It is shown that the technique described allows us to get various 3D oscillations by changing the shape of the trajectories of two 2D-projections of the original 3D system.

Efficient decomposition of singularly perturbed systems

Abstract: The problem of the decomposition of singularly perturbed differential systems by the method of integral manifolds is studied and the application of the method to the problems of enzyme kinetics is considered.

Fractional advection–diffusion equation with memory and Robin-type boundary condition

Abstract: The one-dimensional fractional advection–diffusion equation with Robin-type boundary conditions is studied by using the Laplace and finite sine-cosine Fourier transforms. The mathematical model with memory is developed by employing the generalized Fick’s law with time-fractional Caputo derivative. The influence of the fractional parameter (the non-local effects) on the solute concentration is studied. It is found that solute concentration can be minimized by decreasing the memory parameter. Also, it is found that, at small values of time the ordinary model leads to minimum concentration, while at large values of the time the fractional model is recommended.

Canards Existence in the Hindmarsh–Rose Model

Abstract: In two previous papers, we have proposed a new method for proving the existence of “canard solutions” on one hand for three- and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand, for four-dimensional singularly perturbed systems with two fast variables; see [4, 5]. The aim of this work is to extend this method, which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh–Rose model.

A model in continuous time and space to study economic migration

Abstract: Population movements modify the environment, land-use, and shape the landscape through urbanization. Furthermore, migration has become one of the most relevant determinants of global human health and social development. The objective of this paper is to provide a framework to understand the economic and natural factors responsible for migration and population agglomerations and their environmental consequences. In this regard, we develop an economic model in continuous time and space adapting Hotelling’s migration law to make individuals react to possible improvements of their welfare. First we show that there is solution to this spatial-dynamic problem. Then, we illustrate the properties of the economy and the associated population dynamics through numerical simulations.

Modeling of excitatory amino acid transporters and clearance of synaptic cleft on millisecond time scale

Abstract: Excitatory Amino Acid Transporters (EAATs) operate over wide time scales in the brain. They maintain low ambient concentrations of the primary excitatory amino acid neurotransmitter glutamate, but they also seem to play a significant role in clearing glutamate from the synaptic cleft in the millisecond time-scale process of chemical communication that occurs between neurons. The detailed kinetic mechanisms underlying glutamate uptake and clearance remain incompletely understood. In this work we used a combination of methods to model EAAT kinetics and gain insight into the impact of transport on glutamate dynamics in a general sense. We derive reliable estimates of the turnover rates of the three major EAAT subtypes expressed in the mammalian cerebral cortex. Previous studies have provided transporter kinetic estimates that vary over an order of magnitude. The values obtained in this study are consistent with estimates that suggest the unitary transporter rates are approximately 20-fold slower than the time course of glutamate in the synapse. A combined diffusion/transport model provides a possible mechanism for the apparent discrepancy.

An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

Abstract: The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝn +1 domain Ω. The piecewise regular boundary of Ω is defined as the union ∂ Ω = Γ1 ∪ Γ0 ∪ Σ, where Γ1 and Γ0 are disjoint, regular, and n -dimensional surfaces. Cauchy boundary data is given in Γ0 , and null Dirichlet data in Σ, while no data is given in Γ1 . This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ0 corresponding to an harmonic function in C 2 (Ω) ∩H1 (Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in the L 2 -norm from the measured Cauchy data to the subset of admissible data characterized by given a priori information, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

Ionic parameters identification of an inverse problem of strongly coupled PDE's system in cardiac electrophysiology using Carleman estimates

Abstract: In this paper, we consider an inverse problem of determining multiple ionic parameters of a 2 × 2 strongly coupled parabolic-elliptic reaction-diffusion system arising in cardiac electrophysiology modelling. We use the bidomain model coupled to an ODE system and we consider a general formalism of physiologicaly-detailed cellular membrane models to describe the ionic exchanges at the microscopic level. Our main result is the uniqueness and a Lipschitz stability estimate of the ion channels con-ductance parameters of the model using subboundary observations over an interval of time. The key ingredients are a global Carleman-type estimate with a suitable observations acting on a part of the boundary.