Ronald B. Guenther

An analytical solution to the solute transport equation with rate‐limited desorption and decay

An analytical solution is derived for the advection-dispersion equation with rate-limited desorption and first-order decay, using an eigenfunction integral equations method. The model equations represent one-dimensional solute transport in a homogeneous isotropic porous medium where the porous medium is saturated with the aqueous solution. The flow field is uniform. Rate-limited desorption is described as a first-order process where the rate is proportional to the difference in concentration between the sorbed phase and the aqueous phase. The solution was verified for the limiting case of equilibrium desorption using the solution of van Genuchten and Alves (1982). Example calculations are presented to show the effect of the desorption rate, decay rate, and distribution coefficient on the rate of contaminant removal from both the aqueous and sorbed phases of a groundwater aquifer. The solution quantifies the expected results, where the larger the desorption and decay rate and the smaller the distribution coefficient, the faster the rate of contaminant removal from the aqueous and sorbed phases.

Boundary value problems on infinite intervals and semiconductor devices

The nonlinear differential equation y″ = f(x, y, y′) , 0 ⩽ x f satisfying Bernstein type growth conditions. We also examine an important application which occurs in the theory of semiconductor devices.

Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

Equations of motion and continuity for fluid flow in a porous medium

SummaryThis paper is devoted to a rigorous derivation of the fundamental laws governing the macroscopic flow of fluids in a porous medium. The derivation is given within the framework of classical continuum mechanics and the resulting equations contain the continuity equation and the Euler equations of motion of hydrodynamics as well as Darcys law as special cases.ZusammenfassungDiese Arbeit behandelt die strenge Ableitung der Fundamentalsätze für makroskopische Strömungen in einem porösen Medium. Die Ableitung wird innerhalb der klassischen Mechanik der Kontinua gegeben, und die daraus gewonnenen Gleichungen enthalten die Kontinuitätsgeichung und die Eulerschen Bewegungsgleichungen der Hydrodynamik und das Darcysche Gesetz als Spezialfälle.

First Noether-type theorem for the generalized variational principle of Herglotz

In this paper we formulate and prove a theorem, which provides the conserved quantities of a system described by the generalized variational principle of Herglotz. This new theorem contains as a special case the classical first Noether theorem. It reduces to it when the generalized variational principle of Herglotz reduces to the classical variational principle. Several examples for applications to physics are given.

Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem

This paper extends the generalized variational principle of Herglotz to one with several independent variables and derives the corresponding generalized Euler–Lagrange equations. The extended principle contains the classical variational principle with several independent variables and the variational principle of Herglotz as special cases. A first Noether-type theorem is proven for the new variational principle, which gives the conserved quantities corresponding to symmetries of the associated functional. This theorem contains the classical first Noether theorem as a special case. As examples for applications we calculate a conserved quantity for the damped nonlinear Klein–Gordon equation and we show that the equations which describe the propagation of electromagnetic fields in a conductive medium can be derived from the generalized variational principle of Herglotz (but not from a classical variational principle).

Remarks on parameter identification. I

SummaryIn this paper a method for constructing a spatially varying diffusion coefficient for a parabolic, partial differential equation is given. This function is obtained as the limit of a sequence of functions which are obtained by solving a sequence of finite dimensional optimization problems.

Elastic equations for a cylindrical section of a tree

Considering a cylindrical section of a tree subjected to loads independent of x3 as a relaxed Saint-Venant’s problem, it was shown that plane sections remain plane. Since plane sections remain plane, the displacement equations for the neutral fiber derived using either the relaxed Saint-Venant’s problem or elementary beam theory are equivalent. The stresses in the plane of the transverse cross-section were found to equal to zero. Therefore, it is appropriate to use elementary beam theory to estimate the three-dimensional stress functions when the wood is considered to be homogeneous. In addition the three-dimensional displacement equations allow the required elastic coefficients in cylindrical coordinates to be measured from full size samples. 2002 Elsevier Science Ltd. All rights reserved.

The Diagnosis of Breast Cancer in Mammograms by the Evaluation of Density Patterns

Standard mammograms from 33 patients with surgically proved adenocarcinoma or fibrocystic disease were analyzed with a scanning microdensitometer and computer. A quickly computable number called the linear mass ratio is introduced. This simple ratio discriminated correctly between the 16 adenocarcinomas and 17 fibrocystic lesions of the study, all cases in which diagnosis had required biopsy.

A free boundary value problem for water invading an unsaturated medium

We discuss a mathematical model arising in the filtration of a fluid through a porous medium. The model leads to a free boundary value problem whose governing equation depends on the retention function. A numerical approximation by means of finite elements is used to obtain an existence and uniqueness theorem along with an error estimate for a linear retention function.ZusammenfassungWir diskutieren ein mathematisches Modell für die Filterung einer Flüssigkeit durch ein poröses Medium. Das Modell führt auf ein freies Randwertproblem, dessen Gleichung von der Retentionsfunktion abhängt. Mit Hilfe einer numerischen Finite-Elemente-Approximation werden ein Existenz- und Eindeutigkeitssatz sowie Fehlerabschätzungen für den Fall einer linearen Retentionsfunktion hergeleitet.