Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations

In this work, we investigate existence and uniqueness of solutions for a class of nonlinear multi-point boundary value problems for fractional differential equations. Our analysis relies on the Schauder fixed point theorem and the Banach contraction principle.     

The salting-out effect and phase separation in aqueous solutions of tri-sodium citrate and 1-butyl-3-methylimidazolium bromide

The aim of this work is to obtain further evidence about the salting-out effect produced by the addition of tri-sodium citrate to aqueous solutions of water miscible ionic liquid 1-butyl-3-methylimidazolium bromide ([C₄mim][Br]) by evaluating the effect of tri-sodium citrate on the thermodynamic properties of aqueous solutions of this ionic liquid. Experimental measurements of density and sound velocity at different temperatures ranging from (288.15 to 308.15) K, the refractive index at 308.15 K and the liquid–liquid phase diagram at different temperatures ranging from (288.15 to 338.15) K for aqueous solutions containing 1-butyl-3-methylimidazolium bromide ([C₄mim][Br]) and tri-sodium citrate (Na₃Cit) are taken. The apparent molar volume of transfer of [C₄mim][Br] from water to aqueous solutions of Na₃Cit have positive values and it increases by increasing salt molality. Although at high IL molality, the apparent molar isentropic compressibility shows similar behaviour with that of the apparent molar volume. However at low concentrations of IL, the apparent molar isentropic compressibility of transfer of [C₄mim][Br] from water to aqueous solutions of Na₃Cit have negative values. The effects of temperature and the addition of Na₃Cit and [C₄mim][Br] on the liquid–liquid phase diagram of the investigated system have been studied. It was found that an increase in temperature caused the expansion of the one-phase region. The presence of Na₃Cit triggers a salting-out effect, leading to significant upward shifts of the liquid–liquid de-mixing temperatures of the system. The effect of temperature on the phase-forming ability in the system investigated has been studied based on a salting-out coefficient obtained from fitting the binodal values to a Setschenow-type equation for each temperature. Based on cloud point values, the energetics of the clouding process have been estimated and it was found that both of entropy and enthalpy are the driving forces for biphasic formation.     

Measurement of water activities of alanine + tri-potassium citrate + water system at temperatures between 293.15 and 313 K—Experimental and modeling

Water activity measurements by isopiestic method have been carried out on the aqueous solutions of alanine + tri-potassium citrate (K₃Cit) over a range of temperatures at atmospheric pressure. From these measurements, values of the vapor pressure of solutions were determined. The effect of temperature on the vapor–liquid equilibrium of alanine + K₃Cit + H₂O systems has been studied. The experimental water activities have been correlated successfully with the segment-based local composition Wilson and NRTL models. The agreement between the correlations and the experimental data is good.     

A comparison study between sodium dodecyl sulfate and sodium dodecyl sulfonate with respect to the thermodynamic properties,micellization, and interaction with poly(ethylene glycol) in aqueous solutions

The density, sound velocity, and conductivity measurements were performed on aqueous solutions of sodium dodecyl sulfate (C₁₂H₂₅SO₄Na) or sodium dodecyl sulfonate (C₁₂H₂₅SO₃Na) in the absence and presence of poly(ethylene glycol) (PEG) at different temperatures. Changes in the apparent molar volumes and isentropic compressibilities upon micellization were derived using a pseudophase-transition approach and the infinite dilution apparent molar properties of the monomer and micellar form of C₁₂H₂₅SO₄Na and C₁₂H₂₅SO₃Na were determined. Variations of the critical micelle concentrations (CMCs) of both surfactants in the solutions investigated with temperature were obtained from which thermodynamic parameters of micellization were estimated. It was found that at low temperature the micelle formation process is endothermic and therefore, this process must be entropically driven. However, upon increasing the temperature, the enthalpic factor becomes more significant and, at temperatures higher than 303.15 K the micellization is enthalpy driven. The interactions between C₁₂H₂₅SO₄Na/C₁₂H₂₅SO₃Na and PEG were studied and it was found that sodium alkyl sulfonates were seen to interact more weakly than their sulfate analogues.     

Existence results to positive solutions of fractional BVP with $${\varvec{}}$$-derivatives

This paper investigates the existence and uniqueness of positive and nondecreasing solution for nonlinear boundary value problem with fractional q-derivative

$$\begin{aligned}&D_{q}^{\alpha }u(t)+f(t,u(t))=0, \quad {0<t<1, ~3<\alpha \le 4,} \\&u(0)= D_{q}u(0)=D_{q}^{2}u(0)=0, \quad D_{q}^{2}u(1)=\beta D_{q}^{2}u(\eta ), \end{aligned}$$

where \(D_{q}^{\alpha }\) denotes the Riemann–Liouville q-derivative of order \(\alpha \), \(0<\eta <1\) and \(1-\beta \eta ^{\alpha -3}>0\). Our analysis relies a fixed point theorem in partially ordered sets. An example to illustrate our results is given.

     

Thermodynamics of aqueous solutions of poly ethylene glycol di-methyl ethers in the presence or absence of ammonium phosphate salts

Precise measurements of density and sound velocity at different temperatures ranging from 283.15 to 308.15 K for solutions of PEGDME250, PEGDME500 and PEGDME2000 in water and of PEGDME500 in aqueous solutions of 0.500 mol kg⁻¹ ammonium di-hydrogen phosphate ((NH₄)H₂PO₄) and di-ammonium hydrogen phosphate ((NH₄)₂HPO₄), binodal curves at temperature ranges 293.15-318.15 K for the aqueous PEGDME500 + (NH₄)₂HPO₄, PEGDME500 + (NH₄)₃PO₄, PEGDME2000 + (NH₄)H₂PO₄, PEGDME2000 + (NH₄)₂HPO₄, PEGDME2000 + (NH₄)₃PO₄ and PPG400 + (NH₄)₂HPO₄ two-phase systems, and liquid-liquid equilibrium data at temperature ranges 298.15-318.15 K for the aqueous PEGDME500 + (NH₄)₂HPO₄ and PEGDME2000 + (NH₄)₂HPO₄ two-phase systems have been taken. From the experimental density and sound velocity data, the apparent specific volume, excess specific volume, isentropic compressibility and isentropic compressibility deviation values have been determined and the effect of temperature, charge on the anion of electrolytes and molar mass of PEGDME on the volumetric and compressibility properties of the investigated polymer solutions as well as on the salting-out effect of PEGDMEs produced by ammonium phosphate salts has been studied.     

Extension of the electrolyte NRTL and Wilson models for correlation of viscosity of strong electrolyte solutions at different temperatures

The electrolyte NRTL model [C.C. Chen, L.B. Evans, AIChE J. 32 (1986) 444–454] and electrolyte Wilson model [E. Zhao, M. Yu, R.E. Sauvé, M. Khoshkbarchi, Fluid Phase Equilibr. 173 (2000) 161–175] have been extended for the representation of the dynamic viscosity of strong electrolyte solutions. The models are based on Eyring's absolute rate theory and the electrolyte NRTL and Wilson models for calculating the excess Gibbs energy of activation of the viscous flow. The utility of the models is demonstrated with a successful representation of the viscosity of several electrolyte solutions at different temperatures. The results show that, the model is valid for the whole range of salt concentration and it is reliable for correlation of the viscosity of electrolyte solutions at different temperatures by only four adjustable parameters per binary system.     

Generalized approximation method and a thin film flow of a third grade fluid on a moving belt

We develop a generalized approximation method (GAM) to obtain a solution of a thin film flow of a third grade fluid on a moving belt. The GAM generates a monotone sequence of solutions of linear problems. The sequence of solutions of linear problems converges monotonically and rapidly to a solution of the original nonlinear problem. We present some numerical simulations to illustrate and confirm our results.