On stochastic differential equations and equilibrium distribution: a conditional moment approach

In the recent past there have been several attempts to obtain the equilibrium distribution of multiple populations and their moments in the context of some biological or ecological processes (e.g., Matis and Kiffe in Biometrics 52:155, 1996; Matis and Kiffe in Environ Ecol Stat 9:237, 2002; Renshaw in J Math Appl Med Biol, 15:1, 1998). In particular, the method of cumulant truncation (Matis and Kiffe in Biometrics 52:155, 1996) is a pioneering work in this field. However it requires solving a large number of cumulant functions even in the case of two simultaneous differential equations. Besides the solutions are approximate and depend on the precision of the software. Renshaw (Math Biosci 168:57, 2000) provided a nice extension of the univariate truncated saddle point procedure to multivariate scenarios. But this approach involves a multivariate Newton-Raphson type iterative algorithm whose performance and convergence are critically dependent on the choice of the initial values. In the present paper we propose a new and simple approach to obtain the equilibrium distribution of populations and their conditional moments in a system of differential equations of any dimension. Our proposed method, which is a natural extension of the classical variational matrix approach, has several advantages which are discussed in detail in the paper; among other things it includes the derivation of additional conditions which can be interpreted as environmental surrogates.     

Cu(II) removal using green adsorbents: kinetic modeling and plant scale-up design

Cu(II) adsorption in continuous column using green adsorbents like peanut and almond shell was investigated. Fourier transform infrared (FTIR) spectroscopy, Brunaer-Emmett-Teller (BET) analysis, scanning electron microscopy (SEM), and Point of Zero charge (pH_pzc) determination have been used for characterization of the adsorbents. Experiments were conducted at various operating conditions to calculate the adsorption capacity of the adsorbents. Adsorption studies signify that both the adsorbents have good adsorptive capacity for Cu(II) ion. Equilibrium of adsorption was described using Langmuir isotherm and the highest q_max value for both the adsorbent were obtained at an operating condition of 20 ml/min flow rate, 15 mg/L influent Cu(II) concentration, and 7 cm bed depth. Regeneration of both the adsorbents suggests that these adsorbents can be used several times for Cu(II) removal. Seven different kinetic models were tested among which the modified dose response model was fitted well for peanut shell and the Thomas model was fitted well for almond shell. These fitted models were further used for scale-up design. Regeneration studies show that peanut shell and almond shell are useful up to the fifth adsorption cycle. Application of these adsorbents with industrial effluent was also reported. This study reveals that peanut and almond shells can be used for Cu(II) removal for industrial wastewater.