Search

This paper presents a nonlinear reaction-diffusion-fluid system that simulates radiofrequency ablation within cardiac tissue. The model conveys the dynamic evolution of temperature and electric potential in both the fluid and solid regions, along with the evolution of velocity within the solid region. By formulating the system that describes the phenomena across the entire domain, encompassing both solid and fluid phases, we proceed to an analysis of well-posedness, considering a broad class of right-hand side terms. The system involves parameters such as heat conductivity, kinematic viscosity, and electrical conductivity, all of which exhibit nonlinearity contingent upon the temperature variable. The mathematical analysis extends to establishing the existence of a global solution, employing the Faedo-Galerkin method in a three-dimensional space. To enhance the practical applicability of our theoretical results, we complement our study with a series of numerical experiments. We implement the discrete system using the finite element method for spatial discretization and an Euler scheme for temporal discretization. Nonlinear parameters are linearized through decoupling systems, as introduced in our continuous analysis. These experiments are conducted to demonstrate and validate the theoretical findings we have established.

A nonlinear cross-diffusion--fluid system with chemical terms describing the dynamics of predator-prey living in a Newtonian fluid is proposed in this paper. The existence of a weak solution for the proposed macro-scale system is proved based on the Schauder fixed-point theory, a priori estimates, and compactness arguments. The proposed system is derived from the underlying description delivered by a kinetic-fluid theory model by a multiscale approach. Finally, we discuss the computational results for the proposed macro-scale system in two-dimensional space.

This paper deals with the mathematical analysis and numerical simulation of a new nonlinear ablation system modeling radiofrequency ablation phenomena in cardiac tissue, {which incorporates the effects of blood flow on the heat generated when ablation by radiofrequency. The model also considers the effects of viscous energy dissipation. It consists of a coupled thermistor problem and the incompressible Navier--Stokes equations that describe the evolution of temperature, velocity and potential in cardiac tissue.} In addition to Faedo--Galerkin method, we use Schauder's fixed-point theory to prove the existence of the weak solutions in two- and three-dimensional space. Moreover, we prove the uniqueness of the solution under some additional conditions on the data and the solution. Finally, we discuss some numerical results for the validation of the proposed model using the finite element method.

This paper is devoted to the mathematical analysis of a nonlinear stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noises. Precisely, we detail the proof of the existence of weak martingale solutions by Faedo-Galerkin approximations and the stochastic compactness method. We prove the nonnegativity of solutions by a stochastic adaptation of the Stampacchia approach. Finally, we prove the uniqueness of the solution via duality technique.

This paper deals with the modeling and numerical simulations of multilane vehicular traffic according to the discrete kinetic theory approach. The nonlinear additive interactions and external actions such as tollgates as well traffic signs are considered and modeled. The well-posedness of the related Cauchy problem for the spatially homogeneous case has been proved by using Banach fixed-point theory. Numerical simulations are carried out to validate the improved model in the cases of spatially homogeneous and inhomogeneous problems.

A nonlinear cross-diffusion epidemic with a time-dependent Susceptible-Infected-Recovered-Died system is proposed in this paper. This system is derived from kinetic theory model by multiscale approach, which leads to an equivalent system coupled the microscopic and macroscopic equations. Subsequently, numerical investigations to design asymptotic preserving scheme property is developed and validated by various numerical tests. Finally, the numerical computational results of the proposed system are discussed in two dimensional space using the finite volume method.

In this paper, we propose a time-dependent Susceptible-Exposed-Infectious-Recovered-Died (SEIRD) reaction-diffusion system for the COVID-19 pandemic and we deal with its derivation from a kinetic model. The derivation is obtained by mathematical description delivered at the micro-scale of individuals. Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the kinetic model which couples the microscopic equations with the macroscopic equations. We develop a numerical asymptotic preservation scheme to solve the kinetic model. The proposed approach is validated by various numerical tests where particular attention is paid to the Moroccan situation against the actual pandemic.