Qualitative analysis of the invasion free boundary problem in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the invasion model for multispecies biofilms. This model is based on the continuum approach for biofilm modeling and consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear elliptic partial differential equations describing the substrate trends and a system of semilinear elliptic partial differential equations accounting for the diffusion and reaction of motile species within the biofilm. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. By using the method of characteristics, the partial differential equations constituting the invasion model are converted to Volterra integral equations. Then, the fixed point theorem is used for the uniqueness and existence result. The work is completed with numerical simulations describing the invasion of nitrite oxidizing bacteria in a biofilm initially constituted by ammonium oxidizing bacteria.     

Boundary element solution of the two-dimensional groundwater flow equation with stochastic free surface boundary condition

An innovative approach to the approximate solution of stochastic partial differential equations in groundwater flow is presented. The method uses a formulation of the Ito's lemma in Hilbert spaces to derive partial differential equations satisfying the moments of the solution process. Since the moments equations are deterministic, they could be solved by any analytical or numerical method existing in the literature. This permits the analysis and solution of stochastic partial differential equations occurring in two-dimensional or three-dimensional domains of any geometrical shape. The method is tested for the first time in the present paper through a practical application in a sandy phreatic aquifer at the Chalk River Nuclear Laboratories, Ontario, Canada. The equation solved is the two-dimensional LaPlace equation with a dynamic, randomly perturbed, free surface boundary condition. The moments equations are derived and solved by using the boundary integral equation method. A comparison is made with a previous analytical solution obtained by applying the randomly forced one-dimensional Boussinesq equation, and some observations on modeling procedures are given.     

Space-like Weingarten surfaces in the three-dimensional Minkowski space and their natural partial differential equations

On any space-like Weingarten surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution to the Lund-Regge reduction problem for space-like Weingarten surfaces in Minkowski space. We apply this theory to linear fractional space-like Weingarten surfaces and obtain the natural non-linear partial differential equations describing them. We obtain a characterization of space-like surfaces, whose curvatures satisfy a linear relation, by means of their natural partial differential equations. We obtain the ten natural PDE’s describing all linear fractional space-like Weingarten surfaces.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

Simulation of spreading surfactant on a thin liquid film

The spreading of insoluble surfactant on a thin liquid film is modeled by a pair of nonlinear partial differential equations for the height of the free surface and the surfactant concentration. A numerical method is developed in which the leading edge of the surfactant is tracked. In the absence of higher order regularization the system becomes hyperbolic/degenerate-parabolic, introducing jumps in the height of the free surface and the surfactant concentration gradient. We compare numerical simulations to those of a hybrid Godunov method in which the height equation is treated as a scalar conservation law and a parabolic solver is used for the surfactant equation. We show how the tracking method applies to the full equations with realistic gravity and capillarity terms included, even though the disturbance in the height of the free surface extends beyond the support of the surfactant concentration.     

Timelike surfaces with zero mean curvature in Minkowski 4-space

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this system determine a unique (up to a motion) timelike surface with zero mean curvature so that the given parameters are canonical. We find all timelike surfaces with zero mean curvature in the class of rotational surfaces of Moore type. These examples give rise to a one-parameter family of solutions to the system of natural partial differential equations describing timelike surfaces with zero mean curvature.     

偏微分方程的函数论方法及应用和计算

本文主要介绍了偏微分方程一些边值问题的函数论方法。首先给出了边值问题的适定提法；其次研究了多复变函数、Ｃｌｉｆｆｏｒｄ代数、某类抛物型方程、一些复合型方程组和双曲型方程组各种边值问题的可解性；进而使用一阶椭圆型方程组间断边值问题的结果，解决了渗流理论、空气动力学与弹性力学中提出的若干自由边界问题；最后还讨论了某些椭圆边值问题与拟共形映射的近似解法。从此文可以看出；函数论方法在处理偏微分方程的一些优     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Reduction of PDEs on Unbounded Domains. Application: Unsteady Water-Waves Problem

Summary. For a certain class of partial differential equations in cylindrical domains, we show that all small time-dependent solutions are described by a reduced system of equations on the real line, which contains nonlocal terms. As an application, we investigate the system describing nonlinear water waves travelling on the free surface of an inviscid fluid. Two-dimensional gravity waves are characterized by the parameter λ , the inverse square of the Froude number. For λ close to the critical value λ ₀ =1 , we obtain a reduced system of four nonlocal equations. We show that the terms of lowest order in μ=λ-1 lead to the Korteweg—de Vries equation for the lowest-order approximation of the free surface. Received February 23, 1994; final revision received October 13, 1997; accepted for publication October 16, 1997.     

A system of coupled partial differential equations exhibiting both elevation and depression rogue wave modes

Analytical solutions are obtained for a coupled system of partial differential equations involving hyperbolic differential operators. Oscillatory states are calculated by the Hirota bilinear transformation. Algebraically localized modes are derived by taking a Taylor expansion. Physically these equations will model the dynamics of water waves, where the dependent variable (typically the displacement of the free surface) can exhibit a sudden deviation from an otherwise tranquil background. Such modes are termed ‘rogue waves’ and are associated with ‘extreme and rare events in physics’. Furthermore, elevations, depressions and ‘four-petal’ rogue waves can all be obtained by modifying the input parameters.     

一个药物作用肿瘤生长自由边界问题整体解的存在唯一性

该文研究一个描述药物作用下肿瘤生长的数学模型,这个肿瘤模型是对Jackson模型的一个改进,其数学形式是由一个二阶非线性抛物型方程与两个一阶非线性偏微分方程组耦合而成的自由边界问题.通过运用抛物型方程的L~P理论与一阶偏微分方程的特征方法,并利用Banach不动点定理,证明了该问题存在唯一的整体经典解.     

Traveling waves in excitable media

We consider a two-component reaction-diffusion model that describes the oxygenation of CO molecules on the surface of platinum in the one-dimensional case. The partial differential equations of the model are reduced to a system of ordinary differential equations. We show that the system of partial differential equations with fixed parameter values has a family of autowave solutions running along the spatial axis at various velocities. These solutions are described by some singular attractors and limit cycles of the corresponding period in the system of ordinary differential equations.     

An analytic solution of an oscillatory flow through a porous medium with radiation effect

Analytical solutions for two-dimensional oscillatory flow on free convective-radiation of an incompressible viscous fluid, through a highly porous medium bounded by an infinite vertical plate are reported. The Rosseland diffusion approximation is used to describe the radiation heat flux in the energy equation. The resulting non-linear partial differential equations were transformed into a set of ordinary differential equations using two-term series. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The free-stream velocity of the fluid vibrates about a mean constant value and the surface absorbs the fluid with constant velocity. Expressions for the velocity and the temperature are obtained. To know the physics of the problem analytical results are discussed with the help of graph.     

Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method

Partial differential equations with possibly discontinuous coefficients play an important part in engineering, physics and ecology. In this paper, we will study nonlinear partial differential equations with variable coefficients arising from population models. Generally speaking, it is difficult to analyze the behavior of nonlinear partial differential equations; therefore, we usually rely on the numerical approximation. Currently, there is an increasing interest in designing numerical schemes that preserve energy properties for differential equations. We will design the numerical schemes that preserve discrete energy property and show numerical experiments for a nonlinear partial differential equation with variable coefficients.     

The method of lines for Poisson's equation with nonlinear or free boundary conditions

Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.This work was supported by the U.S. Army Research Office under Grant DA-AG29-76-G-0261     

Free boundary problem for an initial cell layer in multispecies biofilm formation

The initial attached cell layer in multispecies biofilm growth is considered. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The method of characteristics is used to convert the differential system to Volterra integral equations for which an existence and uniqueness theorem is proved. Subsequently, we show that the free boundary is an increasing function of time and biomass concentrations are positive in agreement with the biological process.     

Hopf-Cole transformation to some systems of partial differential equations

In this paper we study a system of nonlinear partial differential equations which we write as a Burgers equation for matrix and use the Hopf-Cole transformation to linearize it. Using this method we solve initial value problem and initial boundary value problems for some systems of parabolic partial differential equations. Also we study an initial value problem for a system of nonlinear partial differential equations of first order which does not have solution in the standard distribution sense and construct an explicit solution in the algebra of generalized functions of Colombeau. Received November 1999     

Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility

In this paper, we show that for a class of nonlinear partial differential equations with arbitrary order the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. The nonlinear wave equation and the Boussinesq equation all serve as examples illustrating this fact.     

Solitary wave solutions to the Tzitzeica type equations obtained by a new efficient approach

The properties of Tzitzeica equations in nonlinear optics have received a great attention of many recent studies. In this work, the so-called generalized exponential rational function method (GERFM) has been applied for finding the analytical solution of two nonlinear partial differential equations type of equations, namely Tzitzeica-Dodd-Bullough and Tzitzeica equation. The proposed method provides a wide range of closed-form travelling solutions leading to a very effective and simply-applied method by means of a symbolic computation system. The method not only provides a general form of solutions with some free parameters but also shows potential application to other types of nonlinear partial differential equations.