Efficient,process-preserving modified Newton's method for solving reactive multicomponent transport problems in porous media

In the context of hydrogeochemical multicomponent transport problems coupled via reaction rates the fully implicit solution of the (discretized) problem can be computationally expensive, but it avoids the consistency errors inherent in the popular operator splitting approaches which separate transport and reaction operators. The efficiency of the globally implicit approach is enhanced however by using Newton's method, analysing the reaction network, and modifying the Jacobian to decouple species equations without deteriorating the performance of the Newton iteration. This is possible in the case of slow kinetics and/or small concentrations. A flexible componentwise sparse matrix linear solver can adapt and change the coupling strategy potentially in every iteration. Numerical results of academical and realistic case studies are presented that demonstrate the potential of the approach. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

We consider the following singularly perturbed boundary-value problem:

Existence of solutions of pseudoconcave inequalities

Tucker's results for systems of linear inequalities are extended to nonlinear systems. Pseudoconcave or homogeneous functions are covered. The proof is performed through a simple, constrained extremum problem. Some applications are also suggested.The author is grateful to Professor M. Morishima and the referee for helpful comments.     

Asymptotic transport models for heat and mass transfer in reactive porous media

We propose an approach for deriving in a rigorous but formal way a family of models of mass and heat transfer in reactive porous media. At a microscopic level we set a model coupling the Boltzmann equation in the gas phase, the heat equation on the solid phase and appropriate interface condititons. An asymptotic expansion leads to a system of coupled diffusion equations where the effective diffusion tensors are defined from the microscopic geometry of the material through auxiliary problems. The ellipticity of the diffusion operator is addressed. To cite this article: P. Charrier, B. Dubroca, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained.     

Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary

This paper deals with the existence and nonexistence of global positive solutions to in ,

and in . Here is a parameter, is a bounded smooth domain. After pointing out the mistakes in Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), 317-326, by N. Wolanski, which claims that, for the ball of , the positive solution exists globally if and only if , we reconsider the same problem in general bounded domain and obtain that every positive solution exists globally if and only if .

     

Existence of periodic solutions

An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 769–784, May, 1997. Translated by V. N. Dubrovsky     

Test solutions of porous media problems

Laminar isothermal fluid flow of two immiscible compressible fluid phases in a porous medium is formulated in terms of four unknown functions ?₁, ?₂, S₁ and S₂ in a pair of partial differential equations, $\text{(}\text{?}\text{?t}\text{)[φ(x, t)S}{}_{\mathrm{i}}\text{?}{}_{\mathrm{i}}\text{] = (}\text{?}\text{?x}\text{){κ(x, t) σ}{}_{\mathrm{i}}\text{(S}{}_{\mathrm{i}}\text{)(}\text{?}\text{?x}\text{)[Φ}{}_{\mathrm{i}}\text{(?}{}_{\mathrm{i}}\text{)]}}$, and a pair of auxiliary relations, S₁ = Γ₁(?₁, ?₂) and S₂ = 1 ? S₁. This system is studied in terms of solutions which are constant along curves, viz., horizontal lines, vertical lines, and parabolas relevant to the parabolic nature of each of the partial differential equations in the system. Such solutions are described, although subject to solution of ordinary differential equations, for use in testing either numerical methods or proposed theorems for the original larger problem.     

一类三种群捕食者-食饵交错扩散模型整体解的存在性

应用能量估计方法和Gagliardo-Nirenberg型不等式证明含一类食饵种群和两类竞争捕食者种群的反应扩散模型整体解的存在性和一致有界性,该模型是带自扩散和交错扩散项的三种群捕食者-食饵模型.     

Pseudomonotone variational inequality problems: Existence of solutions

Necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given. This research was partially done while the author was visiting the University of Chile thanks to the support of an ECOS program.     

Some properties of strong solutions to nonlinear heat and moisture transport in multi-layer porous structures

The present paper deals with mathematical models of heat and moisture transport in layered building envelopes. The study of such processes generates a system of two doubly nonlinear evolution partial differential equations with appropriate initial and boundary conditions. The existence of the strong solution in two dimensions for a (short) time interval is proven. The proof rests on regularity results of elliptic transmission problem for isotropic composite materials.     

Existence and uniqueness of solutions of elastic string with moving ends

A mathematical model for the small vibration of an elastic string is considered. The model takes into account the change of tension due to the movement of the end points of the string. Under the assumptions that the speed of the moving ends be less than the characteristic speed of the equation, the existence and the uniqueness of local weak solution and global strong solution are proved. Copyright © 2004 John Wiley & Sons, Ltd.     

Existence and uniqueness of positive solutions of boundary value problems for coupled systems of singular second-order three-point non-linear differential and difference equations

Using the mixed monotone method we establish the existence and uniqueness of positive solutions for coupled systems of second-order three-point non-linear differential and difference equations. The theorems obtained are very general and complement previously known results.     

Existence and uniqueness of solutions for the magnetohydrodynamic flow of a second grade fluid

In this work, we consider the flow of a second grade fluid in a conducting domain of and in the presence of a magnetic field. When the initial data are of arbitrary size, we prove that the solution of the magnetohydrodynamics problem exists for a small time and is unique. We also show the global existence of solutions for small initial data. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of solutions for gradient systems with application to diffusion problems involving nonconvex energies

In this paper, we establish the existence of solutions for gradient systems of evolution under some type (M) and semi-coerciveness conditions. The main result is applied in order to solve nonlinear diffusion equations involving nonconvex energies.     

Existence of solutions to first-order periodic boundary value problems

This article investigates the existence of solutions to boundary value problems (BVPs) involving systems of first-order ordinary differential equations and two-point, periodic boundary conditions. The methods involve novel differential inequalities and fixed-point theory to yield new theorems guaranteeing the existence of at least one solution.