Parabolic Nonlinear Partial Differential Equations Research Articles

Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology

A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme.

Derivation and analysis of continuum models for crossing pedestrian traffic

In this paper, we study hyperbolic and parabolic nonlinear partial differential equation models, which describe the evolution of two intersecting pedestrian flows. We assume that individuals avoid collisions by sidestepping, which is encoded in the transition rates of the microscopic 2D model. We formally derive the corresponding mean-field models and prove existence of global weak solutions for the parabolic model. Moreover we discuss stability of stationary states for the corresponding one-dimensional model. Furthermore we illustrate the rich dynamics of both systems with numerical simulations.

Modelling the spatiotemporal dynamics of chemovirotherapy cancer treatment

ABSTRACTChemovirotherapy is a combination therapy with chemotherapy and oncolytic viruses. It is gaining more interest and attracting more attention in the clinical setting due to its effective therapy and potential synergistic interactions against cancer. In this paper, we develop and analyse a mathematical model in the form of parabolic non-linear partial differential equations to investigate the spatiotemporal dynamics of tumour cells under chemovirotherapy treatment. The proposed model consists of uninfected and infected tumour cells, a free virus, and a chemotherapeutic drug. The analysis of the model is carried out for both the temporal and spatiotemporal cases. Travelling wave solutions to the spatiotemporal model are used to determine the minimum wave speed of tumour invasion. A sensitivity analysis is performed on the model parameters to establish the key parameters that promote cancer remission during chemovirotherapy treatment. Model analysis of the temporal model suggests that virus burst size and virus infection rate determine the success of the virotherapy treatment, whereas travelling wave solutions to the spatiotemporal model show that tumour diffusivity and growth rate are critical during chemovirotherapy. Simulation results reveal that chemovirotherapy is more effective and a good alternative to either chemotherapy or virotherapy, which is in agreement with the recent experimental studies.

Analysis of a model for the propagation of the ossification front

In this paper, we study a model for the propagation of the ossification front. This is written as a parabolic nonlinear partial differential equation in terms of the density of the osteogenic cells. Its variational formulation leads to a parabolic nonlinear variational equation for which an existence and uniqueness result is proved. Then, we introduce a fully discrete approximation by using the finite element method for the spatial approximation and a Euler scheme to discretize the time derivatives. A priori error estimates are obtained from which, under adequate additional regularity conditions, the linear convergence is derived. Finally, some numerical simulations are shown to demonstrate the accuracy and the influence of the approximation.

Foreword

The study of elliptic and parabolic nonlinear partial differential equation has manifold aspects and can be seen from a number of different perspectives and points of view. The mathematical tools involved can, similarly, be quite different since they may include, for example, functional analysis, calculus of variations, topological techniques, geometric analysis, semigroup theory, numerical methods.  &nbspIt would therefore be hopeless to give a complete picture of the research going on nowadays on these topics. Nevertheless, we tried to collect in the present volume several contributions of some leading scholars in these fields, hoping that they could give some hints of some of the main research lines in the field, identifying in particular some setting in which the elliptic theory is an important clue to analyze the parabolic situation, and viceversa.  &nbspIn the papers collected here, all of which have been anonymously refereed as requested by the high standards of this Journal, several rapidlydeveloping and important topics are discussed and developed.  &nbspAmong them, we mention the following ones:two-phases free boundary problems;qualitative properties of solutions of Lane-Emden-Fowler equations;Hopf fibration and singularly perturbed elliptic equations;biharmonic elliptic boundary value problems;positivity preserving issues in models for clamped plates;Liouville theorems for Hardy-Littlewood-Sobolev systems;regularity of solutions of degenerate elliptic equations;decay properties for degenerate parabolic problems;porous media and fast diffusion equations driven by fractional Laplacians;fine asymptotics of solutions to the fast diffusion equation;well-posedness of nonlinear integral equations with general kernels and asymptotics of the corresponding solutions;local lower and upper bounds for solutions to doubly nonlinear singular parabolic equations;nonlinear initial value problems that model evolution and selection in living systems in connection with kinetic theory;relationships between optimal inequalities and nonlinear flows;generalized solutions, via non-archimedean fields, to equations which may not have solutions in distributional sense;models for oscillations in suspension bridges.  &nbspThe success of this collection depends on the quality of the papers and of the high reputation of the Authors: we are grateful to all the contributors of thepresent volume.

Parallel Multilevel Schwarz and Block Preconditioners for the Bidomain Parabolic-Parabolic and Parabolic-Elliptic Formulations

The aim of this work is to develop parallel multilevel and block preconditioners for the Bidomain model of electrocardiology. The Bidomain model describes the electrical activity of the heart tissue and consists of a system of two parabolic nonlinear partial differential equations (PDEs) of reaction-diffusion type (PP formulation) or alternatively of a system of a parabolic nonlinear PDE and an elliptic linear PDE (PE formulation). In both formulations, the PDEs are coupled with a system of ordinary differential equations, modeling the cellular membrane ionic currents. The first goal of the present study is to construct, analyze, and numerically test a multilevel additive Schwarz preconditioner for the PE formulation of the Bidomain model, extending previous results obtained for the PP formulation. Optimal convergence rate estimates are established and confirmed by 3D numerical test on Linux clusters. The second goal of the present study is to analyze the scalability of multilevel Schwarz block-diagonal and block-factorized preconditioners for both PP and PE formulations of the Bidomain model and to compare them with multilevel Schwarz coupled preconditioners. The 3D parallel numerical tests show that block preconditioners for the PP formulation are not scalable, while they are scalable for the PE formulation, but less efficient than the coupled preconditioners.

THE BOGOMOLNY DECOMPOSITION FOR SYSTEMS OF TWO GENERALIZED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

Using a concept of strong necessary conditions we derive the Bogomolny decomposition for systems of two generalized elliptic and parabolic nonlinear partial differential equations (NPDE) of the second order. The generalization means that the equation coefficients depend on the field variables. According to the Cinquini-Cibrario criteria [18–20] the first type is characterized to be an elliptic, whereas the second one is a parabolic system. As a result we derive conditions for existence of the Bogomolny relationships.

Model of neurotransmitter fast transport in axon terminal of presynaptic neuron

In this paper a methodology of mathematical description of the synthesis, storage and release of the neurotransmitter during the fast synaptic transport is presented. The proposed model is based on the initial and boundary value problem for a parabolic nonlinear partial differential equation (PDE). Presented approach enables to express space and time dependences in the process: rate of vesicular replenishment, gradients of vesicular concentration and, through the boundary conditions, the location of docking and release sites. The model should be a good starting point for future numerical simulations since it is based on thoroughly studied parabolic equation. In the article classical and variational formulation of the problem is presented and the unique solution is shown to exist. The model is referred to the model based on ordinary differential equations (ODEs), created by Aristizabal and Glavinovic (AG model). It is shown that, under some assumptions, AG model is a special case of the introduced one.

The local mass transfer coefficient in spherical film flow by the moving finite element method

A mathematical model has been developed for a differential wetted sphere contactor in which a gas–liquid reaction occurs. The mathematical model given by the parabolic non-linear partial differential equation was solved numerically by the moving finite element method (MFEM) using the Lagrange polynomial approximations of arbitrary degree in each of the finite elements, which the interior nodes optimised as in the orthogonal collocation method. A new approach attaining the local physical mass transfer coefficient near the gas–liquid interface has been demonstrated. The novelty of that approach is that the spherical liquid film exerts a bounding influence on the rate of absorption. The validity of the concept as well as the quantitative predictability of the theoretical analysis is demonstrated by experimentation on the carbon dioxide-water system. Direct comparisons of the local physical mass transfer coefficient obtained with the MFEM and the asymptotic analytical solutions from the literature are presented here.

Upscaling of Vertical Unsaturated Flow Model under Infiltration Condition

A new stochastic model for one-dimensional unsaturated water flow is proposed with focus on its probabilistic structure resulting from random variations in saturated hydraulic conductivity. The newly developed model has the form of a Fokker-Planck equation, and its validity is investigated under different stochastic saturated hydraulic conductivity fields. The point-scale Richards equation for soil water flow, which is a parabolic nonlinear partial differential equation (PDE), is first converted into a simplified ordinary differential equation (ODE) using a depth-integrated scheme. This nonlinear ODE is further converted into a linear PDE using the stochastic Liouville equation. Finally, the upscale conservation equation is obtained using the cumulant expansion method. When compared with Monte Carlo simulations, this model yields good agreements. In particular, this upscale model can reproduce the vertical profile of mean soil-water content very well. Besides, the results from model application show that ...

Nonlinear equations for a thin beam

Finite oscillations of a thin beam are analyzed, taking into account large deflections and rotations with small strains. It is shown that, depending on the assumptions, problems for an elastic and viscoelastic beam lead to solving hyperbolic or parabolic nonlinear partial differential equations. As an example, a cantilever beam is considered. The solutions for nonlinear vibrations are obtained by means of the Galerkin method, and numerical results are presented graphically for four material models.