Inhomogeneous Fractional Diffusion Equations

2000 Mathematics Subject Classification: Primary 26A33; Secondary 35S10, 86A05

Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary

Variationelle Zeitdiskretisierungen höherer Ordnung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten

Wir betrachten zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängi- gen Gebieten, wobei die Bewegung des Gebietsrandes bekannt ist. Die zeitliche Entwicklung des Gebietes wird durch die ALE-Formulierung behandelt, die die Nachteile der klassischen Euler- und Lagrange-Betrachtungsweisen behebt. Die Position des Randes und seine Geschwindigkeit werden dabei so in das Gebietsinnere fortgesetzt, dass starke Gitterdeformationen verhindert werden. Als Zeitdiskretisierungen höherer Ordnung werden stetige Galerkin-Petrov-Verfahren (cGP) und unstetige Galerkin-Verfahren (dG) auf Probleme in zeitabhängigen Gebieten angewendet. Weiterhin werden das C 1 -stetige Galerkin-Petrov-Verfahren und das C 0 -stetige Galerkin- Verfahren vorgestellt. Deren Lösungen lassen sich auch in zeitabhängigen Gebieten durch ein einfaches einheitliches Postprocessing aus der Lösung des cGP-Problems bzw. dG-Problems erhalten. Für Problemstellungen in festen Gebieten und mit zeitlich konstanten Konvektions- und Reaktionstermen werden Stabilitätsresultate sowie optimale Fehlerabschätzungen für die nachbereiteten Lösungen der cGP-Verfahren und der dG-Verfahren angegeben. Für zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten präsentieren wir konservative und nicht-konservative Formulierungen, wobei eine besondere Aufmerksamkeit der Behandlung der Zeitableitung und der Gittergeschwindigkeit gilt. Stabilität und optimale Fehlerschätzungen für die in der Zeit semi-diskretisierten konservativen und nicht-konservativen Formulierungen werden vorgestellt. Abschließend wird das volldiskretisierte Problem betrachtet, wobei eine Finite-Elemente-Methode zur Ortsdiskretisierung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten im ALE-Rahmen einbezogen wurde. Darüber hinaus wird eine lokale Projektionsstabilisierung (LPS) eingesetzt, um der Konvektionsdominanz Rechnung zu tragen. Weiterhin wird numerisch untersucht, wie sich die Approximation der Gebietsgeschwindigkeit auf die Genauigkeit der Zeitdiskretisierungsverfahren auswirkt.

Root Water Extraction under Combined Water and Osmotic Stress

Using a numerical implicit model for root water extraction by a single root in a symmetric radial flow problem, based on the Richards equation and the combined convection-dispersion equation, we investigated some aspects of the response of root water uptake to combined water and osmotic stress. The model implicitly incorporates the effect of simultaneous pressure head and osmotic head on root water uptake, and does not require additional assumptions (additive or multiplicative) to derive the combined effect of water and salt stress. Simulation results showed that relative transpiration equals relative matric flux potential, which is defined as the matric flux potential calculated with an osmotic pressure head-dependent lower bound of integration, divided by the matric flux potential at the onset of limiting hydraulic conditions. In the falling rate phase, the osmotic head near the root surface was shown to increase in time due to decreasing root water extraction rates, causing a more gradual decline of relative transpiration than with water stress alone. Results furthermore show that osmotic stress effects on uptake depend on pressure head or water content, allowing a refinement of the approach in which fixed reduction factors based on the electrical conductivity of the saturated soil solution extract are used. One of the consequences is that osmotic stress is predicted to occur in situations not predicted by the saturation extract analysis approach. It is also shown that this way of combining salinity and water as stressors yields results that are different from a purely multiplicative approach. An analytical steady state solution is presented to calculate the solute content at the root surface, and compared with the outputs of the numerical model. Using the analytical solution, a method has been developed to estimate relative transpiration as a function of system parameters, which are often already used in vadose zone models: potential transpiration rate, root length density, minimum root surface pressure head, and soil theta-h and K-h functions.

Turbulence in Hydraulic Jumps: Experimental Measurements

A hydraulic jump is the transition from a supercritical open channel flow to a subcritical regime. It is characterised by a highly turbulent flow with macro-scale vortices, some kinetic energy dissipation and a bubbly two-phase flow structure. New air-water flow measurements were performed in hydraulic jump flows for a range of inflow Froude numbers. The experiments were conducted in a large-size facility using two types of phase-detection intrusive probes: i.e., single-tip and double-tip conductivity probes. These were complemented by some measurements of free-surface fluctuations using ultrasonic displacement meters. The present study was focused on the turbulence characteristics of hydraulic jumps with partially-developed inflow conditions. The void fraction measurements showed the presence of an advective diffusion shear layer in which the void fractions profiles matched closely an analytical solution of the advective diffusion equation for air bubbles. The present results highlighted some influence of the inflow Froude number onto the air bubble entrainment process. At the largest Froude numbers, the advected air bubbles were more thoroughly dispersed vertically, and larger amount of air bubbles were detected in the turbulent shear layer. In the air-water mixing layer, the maximum void fraction and bubble count rate data showed some longitudinal decay function in the flow direction. Such trends were previously reported in the literature. The measurements of interfacial velocity and turbulence level distributions provided new information on the turbulent velocity field in the highly-aerated shear region. The present data suggested some longitudinal decay of the turbulence intensity. The velocity profiles tended to follow a wall jet flow pattern. The air–water turbulent time and length scales were deduced from some auto- and cross-correlation analyses based upon the method of CHANSON (2006,2007). The results provided the integral turbulent time and length scales of the eddy structures advecting the air bubbles in the developing shear layer. The experimental data showed that the auto-correlation time scale Txx was larger than the transverse cross-correlation time scale Txz. The integral turbulence length scale Lxz was a function of the inflow conditions, of the streamwise position (x-x1)/d1 and vertical elevation y/d1. Herein the dimensionless integral turbulent length scale Lxz/d1 was closely related to the inflow depth: i.e., Lxz/d1 = 0.2 to 0.8, with Lxz increasing towards the free-surface. The free-surface fluctuations measurements showed large turbulent fluctuations that reflected the dynamic, unsteady structure of the hydraulic jumps. A linear relationship was found between the normalized maximum free-surface fluctuation and the inflow Froude number.

Beach water table fluctuations due to spring-neap tides: moving boundary effects

Tidal water table fluctuations in a coastal aquifer are driven by tides on a moving boundary that varies with the beach slope. One-dimensional models based on the Boussinesq equation are often used to analyse tidal signals in coastal aquifers. The moving boundary condition hinders analytical solutions to even the linearised Boussinesq equation. This paper presents a new perturbation approach to the problem that maintains the simplicity of the linearised one-dimensional Boussinesq model. Our method involves transforming the Boussinesq equation to an ADE (advection-diffusion equation) with an oscillating velocity. The perturbation method is applied to the propagation of spring-neap tides (a bichromatic tidal system with the fundamental frequencies wt and wt) in the aquifer. The results demonstrate analytically, for the first time, that the moving boundary induces interactions between the two primary tidal oscillations, generating a slowly damped water table fluctuation of frequency omega(1) - omega(2), i.e., the spring-neap tidal water table fluctuation. The analytical predictions are found to be consistent with recently published field observations. (C) 2000 Elsevier Science Ltd. All rights reserved.

Evaluation of various pre-treatments for the dehydration of banana and selection of suitable drying models

The thin-layer drying behaviour of bananas in a beat pump dehumidifier dryer was examined. Four pre-treatments (blanching, chilling, freezing and combined blanching and freezing) were applied to the bananas, which were dried at 50 degreesC with an air velocity of 3.1 m s(-1) and with the relative humidity of the inlet air of 10-35%. Three drying models, the simple model, the two-term exponential model and the Page model were examined. All models were evaluated using three statistical measures, correlation coefficient, root means square error, and mean absolute percent error. Moisture diffusivity was calculated based on the diffusion equation for an infinite cylindrical shape using the slope method. The rate of drying was higher for the pre-treatments involving freezing. The sample which was blanched only did not show any improvement in drying rate. In fact, a longer drying time resulted due to water absorption during blanching. There was no change in the rate for the chilled sample compared with the control. While all models closely fitted the drying data, the simple model showed greatest deviation from the experimental results. The two-term exponential model was found to be the best model for describing the drying curves of bananas because its parameters represent better the physical characteristics of the drying process. Moisture diffusivities of bananas were in the range 4.3-13.2 x 10(-10) m(2)s(-1). (C) 2002 Published by Elsevier Science Ltd.

Unsaturated diffusivity functions for concrete derived from NMR images

Deterioration of concrete or reinforcing steel through excessive contaminant concentration is often the result of repeated wetting and drying cycles. At each cycle, the absorption of water carries new contaminants into the unsaturated concrete. Nuclear Magnetic Resonance (NMR) is used with large concrete samples to observe the shape of the wetting profile during a simple one-dimensional wetting process. The absorption of water by dry concrete is modelled by a nonlinear diffusion equation with the unsaturated hydraulic diffusivity being a strongly nonlinear function of the moisture content. Exponential and power functions are used for the hydraulic diffusivity and corresponding solutions of the diffusion equation adequately predict the shape of the experimental wetting profile. The shape parameters, describing the wetting profile, vary little between different blends and are relatively insensitive to subsequent re-wetting experiments allowing universal parameters to be suggested for these concretes.

Anomalous water absorption in porous materials

The absorption of fluid by unsaturated, rigid porous materials may be characterized by the sorptivity. This is a simple parameter to determine and is increasingly being used as a measure of a material's resistance to exposure to fluids (especially moisture and reactive solutes) in aggressive environments. The complete isothermal absorption process is described by a nonlinear diffusion equation, with the hydraulic diffusivity being a strongly nonlinear function of the degree of saturation of the material. This diffusivity can be estimated from the sorptivity test. In a typical test the cumulative absorption is proportional to the square root of time. However, a number of researchers have observed deviation from this behaviour when the infiltrating fluid is water and there is some potential for chemo-mechanical interaction with the material. In that case the current interpretation of the test and estimation of the hydraulic diffusivity is no longer appropriate. Kuntz and Lavallee (2001) discuss the anomalous behaviour and propose a non-Darcian model as a more appropriate physical description. We present an alternative Darcian explanation and theory that retrieves the earlier advantages of the simple sorptivity test in providing parametric information about the material's hydraulic properties and allowing simple predictive formulae for the wetting profile to be generated.

Análise de desempenho de algoritmos de otimização para extração de parâmetros em modelos semicondutores de potência

Apresenta-se nesta tese uma revisão da literatura sobre a modelação de semicondutores de potência baseada na física e posterior análise de desempenho de dois métodos estocásticos, Particle Swarm Optimizaton (PSO) e Simulated Annealing (SA), quando utilizado para identificação eficiente de parâmetros de modelos de dispositivos semicondutores de potência, baseado na física. O conhecimento dos valores destes parâmetros, para cada dispositivo, é fundamental para uma simulação precisa do comportamento dinâmico do semicondutor. Os parâmetros são extraídos passo-a-passo durante simulação transiente e desempenham um papel relevante. Uma outra abordagem interessante nesta tese relaciona-se com o facto de que nos últimos anos, os métodos de modelação para dispositivos de potência têm emergido, com alta precisão e baixo tempo de execução baseado na Equação de Difusão Ambipolar (EDA) para díodos de potência e implementação no MATLAB numa estratégia de optimização formal. A equação da EDA é resolvida numericamente sob várias condições de injeções e o modelo é desenvolvido e implementado como um subcircuito no simulador IsSpice. Larguras de camada de depleção, área total do dispositivo, nível de dopagem, entre outras, são alguns dos parâmetros extraídos do modelo. Extração de parâmetros é uma parte importante de desenvolvimento de modelo. O objectivo de extração de parâmetros e otimização é determinar tais valores de parâmetros de modelo de dispositivo que minimiza as diferenças entre um conjunto de características medidas e resultados obtidos pela simulação de modelo de dispositivo. Este processo de minimização é frequentemente chamado de ajuste de características de modelos para dados de medição. O algoritmo implementado, PSO é uma técnica de heurística de otimização promissora, eficiente e recentemente proposta por Kennedy e Eberhart, baseado no comportamento social. As técnicas propostas são encontradas para serem robustas e capazes de alcançar uma solução que é caracterizada para ser precisa e global. Comparada com algoritmo SA já realizada, o desempenho da técnica proposta tem sido testado utilizando dados experimentais para extrair parâmetros de dispositivos reais das características I-V medidas. Para validar o modelo, comparação entre resultados de modelo desenvolvido com um outro modelo já desenvolvido são apresentados.

A methodology for the adaptation of a PMP at the determination of a PMF

This paper presents a new type of very fine grid hydrological model based on the spatiotemporal repartition of a PMP (Probable Maximum Precipitation) and on the topography. The goal is to estimate the influence of this rain on a PMF (Probable Maximum Flood) on a catchment area in Switzerland. The spatiotemporal distribution of the PMP was realized using six clouds modeled by the advection-diffusion equation. The equation shows the movement of the clouds over the terrain and also gives the evolution of the rain intensity in time. This hydrological modeling is followed by a hydraulic modeling of the surface and subterranean flow, done considering the factors that contribute to the hydrological cycle, such as the infiltration, the resurgence and the snowmelt. These added factors make the developed model closer to reality and also offer flexibility in the initial condition that is added to the factors concerning the PMP, such as the duration of the rain, the speed and direction of the wind. All these initial conditions taken together offer a complete image of the PMF.

Time-Delayed Theory of the Neolithic Transition in Europe

The classical wave-of-advance model of the neolithic transition (i.e., the shift from hunter-gatherer to agricultural economies) is based on Fisher's reaction-diffusion equation. Here we present an extension of Einstein's approach to Fickian diffusion, incorporating reaction terms. On this basis we show that second-order terms in the reaction-diffusion equation, which have been neglected up to now, are not in fact negligible but can lead to important corrections. The resulting time-delayed model agrees quite well with observations

Front dynamics in the presence of spatio-temporal structured noises

Front dynamics modeled by a reaction-diffusion equation are studied under the influence of spatiotemporal structured noises. An effective deterministic model is analytical derived where the noise parameters, intensity, correlation time, and correlation length appear explicitly. The different effects of these parameters are discussed for the Ginzburg-Landau and Schlögl models. We obtain an analytical expression for the front velocity as a function of the noise parameters. Numerical simulation results are in a good agreement with the theoretical predictions.

Front dynamics in turbulent media

A study of a stable front propagating in a turbulent medium is presented. The front is generated through a reaction-diffusion equation, and the turbulent medium is statistically modeled using a Langevin equation. Numerical simulations indicate the presence of two different dynamical regimes. These regimes appear when the turbulent flow either wrinkles a still rather sharp propagating interfase or broadens it. Specific dependences of the propagating velocities on stirring intensities appropriate to each case are found and fitted when possible according to theoretically predicted laws. Different turbulent spectra are considered.