This paper concentrates on employing the -difference equations approach to prove another generating function, extended generating function, Rogers formula and Mehler’s formula for the polynomials , as well as thegenerating functions of Srivastava-Agarwal type. Furthermore, we establish links between the homogeneous -difference equations and transformation formulas.

      This paper presents an alternative method for developing effective embedded optimized Runge-Kutta (RK) algorithms to solve oscillatory problems numerically.   The embedded scheme approach has algebraic orders of 5 and 4. By transforming second-order ordinary differential equations (ODEs) into their first-order counterpart, the suggested approach solves first-order ODEs. The amplification error, phase-lag, and first derivative of the phase-lag are all nil in the embedded pair. The alternative method’s absolute stability is demonstrated. The numerical tests are conducted to demonstrate the effectiveness of the developed approach in comparison to other RK approaches. The alternative approach outperforms the current RK methods

The nanostructured Manganese dioxide/Carbon fiber (CF) composite electrode was prepared galvanostatically using a facile method of anodic electrodeposition by varying the reaction time and MnSO4 concentration of the electrochemical solution. The effects of these parameters on the structures and properties of the prepared electrode were evaluated. For determining the crystal characteristics, morphologies, and topographies of the deposited MnO2 films onto the surfaces of carbon fibers, the X-ray diffraction (XRD), field emission scanning electron microscopy (FESEM), and atomic force microscopy (AFM) techniques were used, respectively. It found that the carbon fibers were coated with γ-MnO2 with a density that increased with increasing the de

The one-dimensional, cylindrical coordinate, non-linear partial differential equation of transient heat conduction through a hollow cylindrical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical
function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant the

The one-dimensional, spherical coordinate, non-linear partial differential equation of transient heat conduction through a hollow spherical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal con

This work is aiming to study and compare the removal of lead (II) from simulated wastewater by activated carbon and bentonite as adsorbents with particle size of 0.32-0.5 mm. A mathematical model was applied to describe the mass transfer kinetic.

The batch experiments were carried out to determine the adsorption isotherm constants for each adsorbent, and five isotherm models were tested to choose the best fit model for the experimental data. The pore, surface diffusion coefficients and mass transfer coefficient were found by fitting the experimental data to a theoretical model. Partial differential equations were used to describe the adsorption in the bulk and solid phases. These equations were simplified and the

The mixed-spin ferrimagnetic Ising system consists of two-dimensional sublattices A and B with spin values and respectively .By used the mean-field approximation MFA of Ising model to find magnetism( ).In order to determined the best stabile magnetism , Gibbs free energy employ a variational method based on the Bogoliubov inequality .The ground-state (Phase diagram) structure of our system can easily be determined at , we find six phases with different spins values depend on the effect of a single-ion anisotropies .these lead to determined the second , first orders transition ,and the tricritical points as well as the compensation phenomenon .

    The  adsorption of cephalexin.H₂O from aqueous solution on attapulgite, bentonite and kaolin has been studied at the human body temperature (37.5ËšC) and at 5, 27, 47ËšC in 0.1M hydrochloric acid (pH 1.2). The value of pH 1.2 has been chosen to simulate the pH of stomach fluid. The clays show the following order: Bentonite > attapulgite > kaolin, for their activity to adsorb cephalexin.H₂O. The charged clay particles can attract molecules either by electrostatic forces, for the molecules of oppositely charged, or by inducing dipole formation in the neutral molecule. The L-shaped adsorption isotherm indicated that drug molecules arrangement in a flat geometry on the clay surface. The