Parabolic PDE Systems Research Articles

Passive Fuzzy Control Design for a Class of Nonlinear Distributed Parameter Systems with Time-varying Delay

This paper is devoted to studying the issue of passive fuzzy controller design for a class of nonlinear distributed parameter systems represented by semi-linear parabolic partial differential equations. The main objective of this paper is to develop two kinds of fuzzy controllers, one is static output feedback controller (SOFC), the other is dynamic output feedback controller (DOFC), which can guarantee both the stability and passivity of the designed closed-loop system. For the purpose of achieving the anticipated target, in this paper, the semi-linear parabolic PDE systems are assumed to be exactly represented by a Takagi-Sugeno (T-S) fuzzy parabolic PDE model. Furthermore, two examples are given to demonstrate the effectiveness of the controller design scheme.

Modified bidomain model with passive periodic heterogeneities

<p style='text-indent:20px;'>In this paper we study how mesoscopic heterogeneities affect electrical signal propagation in cardiac tissue. The standard model used in cardiac electrophysiology is a bidomain model - a system of degenerate parabolic PDEs, coupled with a set of ODEs, representing the ionic behviour of the cardiac cells. We assume that the heterogeneities in the tissue are periodically distributed diffusive regions, that are significantly larger than a cardiac cell. These regions represent the fibrotic tissue, collagen or fat, that is electrically passive. We give a mathematical setting of the model. Using semigroup theory we prove that such model has a uniformly bounded solution. Finally, we use two–scale convergence to find the limit problem that represents the average behviour of the electrical signal in this setting.

An enhanced state estimation for linear parabolic PDE systems with mobile sensors

SummaryThis paper studies an enhanced state estimation problem of distributed parameter processes modeled by a linear parabolic partial differential equation using mobile sensors. The proposed estimation scheme contains a state estimator and the guidance of mobile sensors, where the spatial domain is decomposed into multiple subdomains according to the number of sensors and each sensor is capable of moving within the respective subdomain. The state estimator is desired to make the state estimation error system exponentially stable while providing an performance bound. The mobile sensor guidance is used to enhance the transient performance of the error system. By the Lyapunov direct technique, an integrated design of state estimator and mobile sensor guidance laws is developed in the form of bilinear matrix inequalities (BMIs) to meet the desired design objectives. Moreover, to make the performance bound as small as possible, a suboptimal enhanced state estimation problem is formulated as a BMI optimization one, which can be solved via an iterative linear matrix inequality algorithm. Finally, numerical simulations are given to show the effectiveness of the proposed method.

On the existence–uniqueness and computation of solution of a class of cardiac electric activity models

In this study, we consider a system of coupled PDEs–ODEs which govern the electric activity in heart with a diffusion term modeling the potential in the surrounding tissue and the nonlinear ionic model proposed by the different reduced cell level models. We prove the existence and uniqueness of the bidomain model with the Morris and Lecar ionic model. The global existence of solution is established based on regularization argument using Fedo–Galerkin/compactness approach. The uniqueness of the solution is shown based on Gronwell’s lemma upon some special treatment of nonlinear terms. The computational realization of the monodomain model with different reduced ionic test models is presented. Also, the coupled parabolic PDE and ODE system modeling the cardiac electric activity in a monodomain system representation of cardiac tissue with different ionic models (at cell level), viz. FitzHugh–Nagumo model, Roger–McCulloch model (RMM), Aliev–Panfilov model and Mitchell–Schaeffer model (MSM), is numerically analyzed by Galerkin linear finite elements in space and the backward Euler method in time. Simulations are done using FreeFem++ open source, and the evolution pattern of the solution has been investigated. MSM and RMM models are unconditionally stable with respect to applied stimulus and initial state, although other models are conditionally stable.

Solvability of a Nonlinear Problem Arising in Reactions over Inhomogeneous Surfaces

The existence and uniqueness theorem of classical solutions of a coupled system of nonlinear parabolic PDEs arising in modelling of chemical reactions between two polymeric reactants over inhomogeneous surfaces with nonclassical boundary conditions is proved and the long-time behaviour of the solution is studied.

A multicomponent flow model in deformable porous media

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium.

Stability and Error Estimates of Fully Discrete Schemes for the Brusselator System

Space-time approximations of the Brusselator system of parabolic PDEs are examined. The schemes under consideration are discontinuous (in time) combined with standard conforming finite elements in ...

The robust finite-volume schemes for modeling nonclassical surface reactions

A coupled system of nonlinear parabolic PDEs arising in modeling of surface reactions with piecewise continuous kinetic data is studied. The nonclassic conjugation conditions are used at the surface of the discontinuity of the kinetic data. The finite-volume technique and the backward Euler method are used to approximate the given mathematical model. The monotonicity, conservativity, positivity of the approximations are investigated by applying these finite-volume schemes for simplified subproblems, which inherit main new nonstandard features of the full mathematical model. Some results of numerical experiments are discussed.

An efficient B-spline difference method for solving system of nonlinear parabolic PDEs

This article tries to use of cubic B-spline collocation method in the form of a difference method to solving system of one dimension nonlinear parabolic partial differential equations. This method is characterized by the simplicity of the finite difference method through discussion of various parabolic pde problems, but does not have complex computation of traditional cubic B-spline collocation method. The use of this method leads to a system of algebraic equations which is suitable for computer programming. The stability and convergence of this method is discussed. The obtained numerical solutions indicate that the method is reliable and yields results compatible with the exact solutions. The numerical approximate solutions to the nonlinear parabolic partial differential equations have been computed without transforming the equation and without using the linearization.

Global existence of solutions to a tear film model with locally elevated evaporation rates

Motivated by a model proposed by Peng et al. (2014) for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDEs for the film thickness and salt concentration subject to non-conservative effects representing evaporation. We analytically prove the global existence of solutions to this model with mobility exponents in several different ranges and present numerical simulations that are in agreement with the analytic results. We also numerically capture other interesting dynamics of the model, including finite-time rupture–shock phenomenon due to the instabilities caused by locally elevated evaporation rates, convergence to equilibrium and infinite-time thinning.

Isogeometric overlapping Schwarz preconditioners for the Bidomain reaction–diffusion system

We present and study an overlapping additive Schwarz preconditioner for the isogeometric discretization of a reaction–diffusion system modeling the heart bioelectrical activity, known as the Bidomain model, consisting of a degenerate system of parabolic and elliptic PDE. This model includes intramural fiber rotation, anisotropic conductivity coefficients and is coupled through the reaction term with a system of ODEs, which models the ionic currents of the cellular membrane. The overlapping Schwarz preconditioner is applied with a PCG accelerator to solve the linear system arising at each time step from the isogeometric discretization in space and a semi-implicit adaptive method in time. A theoretical convergence rate analysis shows that the resulting solver is scalable, optimal in the ratio of subdomain/element size and the convergence rate improves with increasing overlap size. Numerical tests in three-dimensional ellipsoidal domains confirm the theoretical estimates and additionally show the robustness with respect to jump discontinuities of the orthotropic conductivity coefficients.

Evolution families and maximal regularity for systems of parabolic equations

In this paper, we prove maximal $L^p$-regularity for a system of parabolic PDEs, where the elliptic operator $A$ has coefficients which depend on time in a measurable way and are continuous in the space variable. The proof is based on operator-theoretic methods and one of the main ingredients in the proof is the construction of an evolution family on weighted $L^q$-spaces.

A non-linear system of parabolic type PDEs for epidemic models

A non-linear system of parabolic type PDEs for epidemic models

Advection control in parabolic PDE systems for competitive populations

This paper investigates the response of two competing species to a given resource using optimal control techniques. We explore the choices of directed movement through controlling the advective coefficients in a system of parabolic partial differential equations. The objective is to maximize the abundance represented by a weighted combination of the two populations while minimizing the 'risk' costs caused by the movements.

Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model

In this paper, we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles, in which the free energy of the system consists of an elastic bending energy, a surface area constraint and a volume constraint. One major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi-discrete level. We develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic PDE system based on the “Invariant Energy Quadratization” approach. In particular, the resulted semi-discretizations lead to linear systems in space with symmetric positive definite operators at each time step, thus can be efficiently solved. In addition, the proposed schemes are rigorously proved to be unconditionally energy stable. Various numerical experiments in 2D and 3D are presented to demonstrate the stability and accuracy of the proposed schemes.

Discrete nonnegativity for nonlinear cooperative parabolic PDE systems with non-monotone coupling

Discrete nonnegativity principles are established for finite element approximations of nonlinear parabolic PDE systems with mixed boundary conditions. Previous results of the authors are extended such that diagonal dominance (or essentially monotonicity) of the nonlinear coupling can be relaxed, allowing to include much more general situations in suitable models.

A fitness-driven cross-diffusion system from population dynamics as a gradient flow

We consider a fitness-driven model of dispersal of N interacting populations, which was previously studied merely in the case N=1. Based on some optimal transport distance recently introduced, we identify the model as a gradient flow in the metric space of Radon measures. We prove existence of global non-negative weak solutions to the corresponding system of parabolic PDEs, which involves degenerate cross-diffusion. Under some additional hypotheses and using a new multicomponent Poincaré–Beckner functional inequality, we show that the solutions converge exponentially to an ideal free distribution in the long time regime.

Dissipativity-based observer design for a class of coupled 1-D semi-linear parabolic PDE systems

In the present paper the dissipativity-based observer design approach is extended to a class of 1-D semi-linear parabolic PDE systems with pointwise in-domain measurement. On the basis of a modal innovation over the set of slow eigenvalues of the linear operator and the consideration of a dissipativity (sector) condition for the nonlinearity an exponential convergence condition for the estimation error is derived in form of a linear operator inequality, which is solved on the basis of an eigenfunction expansion of the linear operator. The approach is illustrated by numerical simulations for a coupled two-state semi-linear PDE system.

Hölder Continuity of Solutions to the Kinematic Dynamo Equations

We study the propagation of regularity of solutions to a three dimensional system of linear parabolic PDE known as the kinematic dynamo equations. The divergence free drift velocity is assumed to be at the critical regularity level with respect to the natural scaling of the equations.

Optimal Tracking for a Divergent-Type Parabolic PDE System in Current Profile Control

We consider the optimal tracking problem for a divergent-type parabolic PDE system, which can be used to model the spatial-temporal evolution of the magnetic diffusion process in a tokamak plasma. With the feedforward trajectories generated by numerical optimization, we derive a linear system to describe the error evolution. The weak variation approach is then extended to cylindrical coordinates to obtain a Riccati-type PDE for feedback kernel synthesis. Then, a finite difference method is used to give numerical solutions for the Riccati-type PDE. The obtained feedback kernel is finally used to simulate the closed-loop system. The simulation results demonstrate that the proposed tracking method can attenuate the tracking error effectively.