On existence and uniqueness of positive solutions to a class of fractional boundary value problems

[EN] The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = 0 , where 2 < α ≤ 3 and D 0 + α is the Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear), the question of uniqueness of the solution is not treated. We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear). 2010 Mathematics Subject Classification: 34B15

Bioremediation of PAHs - Limitations and soultions

Introduction 1.1 Occurrence of polycyclic aromatic hydrocarbons (PAH) in the environment Worldwide industrial and agricultural developments have released a large number of natural and synthetic hazardous compounds into the environment due to careless waste disposal, illegal waste dumping and accidental spills. As a result, there are numerous sites in the world that require cleanup of soils and groundwater. Polycyclic aromatic hydrocarbons (PAHs) are one of the major groups of these contaminants (Da Silva et al., 2003). PAHs constitute a diverse class of organic compounds consisting of two or more aromatic rings with various structural configurations (Prabhu and Phale, 2003). Being a derivative of benzene, PAHs are thermodynamically stable. In addition, these chemicals tend to adhere to particle surfaces, such as soils, because of their low water solubility and strong hydrophobicity, and this results in greater persistence under natural conditions. This persistence coupled with their potential carcinogenicity makes PAHs problematic environmental contaminants (Cerniglia, 1992; Sutherland, 1992). PAHs are widely found in high concentrations at many industrial sites, particularly those associated with petroleum, gas production and wood preserving industries (Wilson and Jones, 1993). 1.2 Remediation technologies Conventional techniques used for the remediation of soil polluted with organic contaminants include excavation of the contaminated soil and disposal to a landfill or capping - containment - of the contaminated areas of a site. These methods have some drawbacks. The first method simply moves the contamination elsewhere and may create significant risks in the excavation, handling and transport of hazardous material. Additionally, it is very difficult and increasingly expensive to find new landfill sites for the final disposal of the material. The cap and containment method is only an interim solution since the contamination remains on site, requiring monitoring and maintenance of the isolation barriers long into the future, with all the associated costs and potential liability. A better approach than these traditional methods is to completely destroy the pollutants, if possible, or transform them into harmless substances. Some technologies that have been used are high-temperature incineration and various types of chemical decomposition (for example, base-catalyzed dechlorination, UV oxidation). However, these methods have significant disadvantages, principally their technological complexity, high cost , and the lack of public acceptance. Bioremediation, on the contrast, is a promising option for the complete removal and destruction of contaminants. 1.3 Bioremediation of PAH contaminated soil & groundwater Bioremediation is the use of living organisms, primarily microorganisms, to degrade or detoxify hazardous wastes into harmless substances such as carbon dioxide, water and cell biomass Most PAHs are biodegradable unter natural conditions (Da Silva et al., 2003; Meysami and Baheri, 2003) and bioremediation for cleanup of PAH wastes has been extensively studied at both laboratory and commercial levels- It has been implemented at a number of contaminated sites, including the cleanup of the Exxon Valdez oil spill in Prince William Sound, Alaska in 1989, the Mega Borg spill off the Texas coast in 1990 and the Burgan Oil Field, Kuwait in 1994 (Purwaningsih, 2002). Different strategies for PAH bioremediation, such as in situ , ex situ or on site bioremediation were developed in recent years. In situ bioremediation is a technique that is applied to soil and groundwater at the site without removing the contaminated soil or groundwater, based on the provision of optimum conditions for microbiological contaminant breakdown.. Ex situ bioremediation of PAHs, on the other hand, is a technique applied to soil and groundwater which has been removed from the site via excavation (soil) or pumping (water). Hazardous contaminants are converted in controlled bioreactors into harmless compounds in an efficient manner. 1.4 Bioavailability of PAH in the subsurface Frequently, PAH contamination in the environment is occurs as contaminants that are sorbed onto soilparticles rather than in phase (NAPL, non aqueous phase liquids). It is known that the biodegradation rate of most PAHs sorbed onto soil is far lower than rates measured in solution cultures of microorganisms with pure solid pollutants (Alexander and Scow, 1989; Hamaker, 1972). It is generally believed that only that fraction of PAHs dissolved in the solution can be metabolized by microorganisms in soil. The amount of contaminant that can be readily taken up and degraded by microorganisms is defined as bioavailability (Bosma et al., 1997; Maier, 2000). Two phenomena have been suggested to cause the low bioavailability of PAHs in soil (Danielsson, 2000). The first one is strong adsorption of the contaminants to the soil constituents which then leads to very slow release rates of contaminants to the aqueous phase. Sorption is often well correlated with soil organic matter content (Means, 1980) and significantly reduces biodegradation (Manilal and Alexander, 1991). The second phenomenon is slow mass transfer of pollutants, such as pore diffusion in the soil aggregates or diffusion in the organic matter in the soil. The complex set of these physical, chemical and biological processes is schematically illustrated in Figure 1. As shown in Figure 1, biodegradation processes are taking place in the soil solution while diffusion processes occur in the narrow pores in and between soil aggregates (Danielsson, 2000). Seemingly contradictory studies can be found in the literature that indicate the rate and final extent of metabolism may be either lower or higher for sorbed PAHs by soil than those for pure PAHs (Van Loosdrecht et al., 1990). These contrasting results demonstrate that the bioavailability of organic contaminants sorbed onto soil is far from being well understood. Besides bioavailability, there are several other factors influencing the rate and extent of biodegradation of PAHs in soil including microbial population characteristics, physical and chemical properties of PAHs and environmental factors (temperature, moisture, pH, degree of contamination). Figure 1: Schematic diagram showing possible rate-limiting processes during bioremediation of hydrophobic organic contaminants in a contaminated soil-water system (not to scale) (Danielsson, 2000). 1.5 Increasing the bioavailability of PAH in soil Attempts to improve the biodegradation of PAHs in soil by increasing their bioavailability include the use of surfactants , solvents or solubility enhancers.. However, introduction of synthetic surfactant may result in the addition of one more pollutant. (Wang and Brusseau, 1993).A study conducted by Mulder et al. showed that the introduction of hydropropyl-ß-cyclodextrin (HPCD), a well-known PAH solubility enhancer, significantly increased the solubilization of PAHs although it did not improve the biodegradation rate of PAHs (Mulder et al., 1998), indicating that further research is required in order to develop a feasible and efficient remediation method. Enhancing the extent of PAHs mass transfer from the soil phase to the liquid might prove an efficient and environmentally low-risk alternative way of addressing the problem of slow PAH biodegradation in soil.

Variable-density jet flows induced by concentrated sources of momentum and energy

The planar and axisymmetric variable-density flows induced in a quiescent gas by a concentrated source of momentum that is simultaneously either a source or a sink of energy are investigated for application to the description of the velocity and temperature far fields in laminar gaseous jets with either large or small values of the initial jet-to-ambient temperature ratio. The source fluxes of momentum and heat are used to construct the characteristic scales of velocity and length in the region where the density differences are of the order of the ambient density, which is slender for the large values of the Reynolds number considered herein. The problem reduces to the integration of the dimensionless boundary-layer conservation equations, giving a solution that depends on the gas transport properties but is otherwise free of parameters. The boundary conditions at the jet exit for integration are obtained by analysing the self-similar flow that appears near the heat source in planar and axisymmetric configurations and also near the heat sink in the planar case. Numerical integrations of the boundary-layer equations with these conditions give solutions that describe accurately the velocity and temperature fields of very hot planar and round jets and also of very cold plane jets in the far field region where the density and temperature differences are comparable to the ambient values. Simple scaling arguments indicate that the point source description does not apply, however, to cold round jets, whose far field region is not large compared with the jet development region, as verified by numerical integrations

Control of a Teleoperation System by State Convergence with Variable Time Delay

In this paper, we propose a novel control scheme for bilateral teleoperation of n degree-of-freedom (DOF) nonlinear robotic systems with time-varying communication delay. We consider that the human operator contains a constant force on the local manipulator. The local and remote manipulators are coupled using state convergence control scheme. By choosing a Lyapunov-Krasovskii functional, we show that the local-remote teleoperation system is asymptotically stable. It is also shown that, in the case of reliable communication protocols, the proposed scheme guarantees that the remote manipulator tracks the delayed trajectory of the local manipulator. The time delay of communication channel is assumed to be unknown and randomly time varying, but the upper bounds of the delay interval and the derivative of the delay are assumed to be known.

Cauchy Problem for Differential Equation with Caputo Derivative

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given.

Operational Rules for a Mixed Operator of the Erdélyi-Kober Type

2000 Mathematics Subject Classification: 26A33 (main), 44A40, 44A35, 33E30, 45J05, 45D05

An Expansion Formula for Fractional Derivatives and its Application

An expansion formula for fractional derivatives given as in form of a series involving function and moments of its k-th derivative is derived. The convergence of the series is proved and an estimate of the reminder is given. The form of the fractional derivative given here is especially suitable in deriving restrictions, in a form of internal variable theory, following from the second law of thermodynamics, when applied to linear viscoelasticity of fractional derivative type.

Discrete Models of Time-Fractional Diffusion in a Potential Well

Mathematics Subject Classification: 26A33, 45K05, 60J60, 60G50, 65N06, 80-99.

Numerical Approximation of a Fractional-In-Space Diffusion Equation, I

2000 Mathematics Subject Classification: 26A33 (primary), 35S15 (secondary)

On Multidimensional Analogue of Marchaud Formula for Fractional Riesz-Type Derivatives in Domains in R^n

2000 Mathematics Subject Classification: 26A33, 42B20

On q–Analogues of Caputo Derivative and Mittag–Leffler Function

Mathematics Subject Classification: 33D60, 33E12, 26A33

Mixed Fractional Integration Operators in Mixed Weighted Hölder Spaces

MSC 2010: 26A33

Nonlinear Time-Fractional Differential Equations in Combustion Science

MSC 2010: 34A08 (main), 34G20, 80A25

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

Solutions of Fractional Diffusion-Wave Equations in Terms of H-functions

MSC 2010: 35R11, 42A38, 26A33, 33E12