An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation

In this paper, an optimization method based on a new class of basis functions, namely generalized polynomials (GPs), is proposed for nonlinear variable-order time fractional diffusion-wave equation. Variable-order time fractional derivative is expressed in the Caputo sense. In the proposed method, solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. In this way, some new operational matrices of the ordinary and fractional derivatives are derived for these basis functions. The residual function and its 2-norm are employed for converting the problem under study to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and effectiveness of the method are demonstrated by solving some numerical examples. The results demonstrate that the proposed method is a powerful algorithm with good accuracy for solving such kind of problems.     

Solving two-dimensional Volterra-Fredholm integral equations of the second kind by using Bernstein polynomials

In this paper, we present a numerical method for solving two-dimensional Volterra-Fredholm integral equations of the second kind (2DV-FK2). Our method is based on approximating unknown function with Bernstein polynomials. We obtain an error bound for this method and employ the method on some numerical tests to show the efficiency of the method.     

Determination of zinc in water samples by flame atomic absorption spectrometry after homogeneous liquid-liquid extraction

In this study, a new and simple homogeneous liquid-liquid extraction (HLLE) method based on a pH-independent phase-separation process was developed using a ternary solvent system [water-tetrabutylammonium ion (TBA ⁺)-chloroform] for the preconcentration of Zn²⁺ ions. A Schiff’s base ligand was used as the chelating agent prior to Zn²⁺ ions extraction. Flame atomic absorption spectrophotometry using acetylene-air flame was used for the quantification of analyte after preconcentration. The phase separation occurred due to ion-pair formation of TBA and perchlorate ion. The sedimented phase was then separated using a 100 μL micro-syringe and diluted to 1.0 mL with ethanol. The sample was introduced into the flame by conventional aspiration. After the optimization of complexation and extraction conditions such as pH 9.0, [ligand] = 1.0 × 10⁻⁵ M, [TBA⁺] = 2.0 × 10⁻² M, 100.0 μL of [CHCl₃] and [CLO₄⁻] = 2.0 × 10⁻² M, a preconcentration factor of 100 was achieved for only 10 mL of the sample. The relative standard deviation was 2.3% (n = 10). The limit of detection was sufficiently low and at ppb level. The proposed method was applied to the extraction and determination of Zn²⁺ in natural water samples with satisfactory results.     

Preconcentration and separation of ultra-trace palladium ion using pyridine-functionalized magnetic nanoparticles

We present a study on the application of magnetic nanoparticles (MNPs) prepared from Fe₃O₄ and functionalized with pyridine as an adsorbent for the solid-phase extraction of trace quantities of Pd(II) ion. The pyridine group was immobilized on the surface of the MNPs by covalent bonding of isonicotinamide. The modified MNPs can be readily separated from an aqueous solution by applying an external magnetic field. Effects of pH, the amount of functionalized MNPs, extraction time, type and quantity of eluent, desorption time, break-through volume and interfering ions on the extraction efficiency were optimized. The amount of Pd(II) was then determined using FAAS. Under the optimized conditions, the detection limit and preconcentration factor are 0.15?μg?L^-1 and 196, respectively, and the relative standard deviation (at 20?μgL^?1; for n?=?10) is 3.7?%. The method had a linear analytical range from 1 to 80?μg?L^-1 and was applied to determine Pd(II) in spiked tape water and soil.

Synthesis and application of a novel magnetic nanosorbent for determination of trace Cd(II), Ni(II), Pb(II), and Zn(II) in environmental samples

In this work for the first time, Fe₃O₄@SiO₂ core–shell nanoparticles functionalized with isatin groups as a magnetic nanosorbent was applied for the simultaneous extraction of trace amounts of cadmium(II), nickel(II), lead(II), and zinc(II). The characterization of this nanosorbent was studied using Fourier transform infrared spectroscopy, scanning electron microscopy, energy-dispersive X-ray spectrometry, X-ray diffraction, vibrating sample magnetometer and thermogravimetric analysis. The effect of several factors such as pH, amount of sorbent, extraction time, type and volume of the eluent, sample volume, sorption capacity, and potentially interfering ions was investigated. In the selected conditions, it was observed that the limits of detection were 0.11 ng mL^?1 for Cd(II), 0.28 ng mL^?1 for Ni(II), 0.47 ng mL^?1 for Pb(II), and 0.21 ng mL^?1 for Zn(II), and the maximum sorption capacity of this suggested magnetic nanosorbent was 120, 112, 100, and 100 mg g^?1 for Cd(II), Ni(II), Pb(II), and Zn(II), respectively. Also, the precision of the method (RSD%) for ten replicate measurements was found 2.5, 2.5, 2.8, and 3.1%, for Cd(II), Ni(II), Pb(II), and Zn(II) ions, respectively. Finally, the suggested procedure was applied for determination of cadmium(II), nickel(II), lead(II), and zinc(II) at trace levels in different water and agricultural products with satisfactory results.     

Synthesis and application of a novel magnetic SBA-15 nanosorbent for heavy metal removal from aqueous solutions

Journal of Sol-Gel Science and Technology - In this work for the first time, magnetic nanoporous SBA-15 functionalized with isatin groups as a magnetic nanosorbent was applied for the simultaneous...