The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

n this work, the adsorption of crystal violet dye from aqueous solution on charcoal and rice husk has been investigated, where the impact of variable factors (contact time; the dosage of adsorbent, pH, temperature, and ionic strength) have been studied. It has been found that charcoal and rice husk have an appropriate adsorption limit with regards to the expulsion of crystal violet dye from fluid arrangements. The harmony adsorption is for all intents and purposes accomplished in 45 min for charcoal and 60 min for rice husk. The amount of crystal violet dye adsorbed (0.4 g of charcoal and 0.5 g of rice husk) increased with an increasing pH and the value of 11 is the best. The effect of temperature on the adsorption process was studied

            In this work, the adsorption of crystal violet dye from aqueous solution on charcoal and rice husk has been investigated, where the impact of variable factors (contact time; the dosage of adsorbent, pH, temperature, and ionic strength) have been studied. It has been found that charcoal and rice husk have an appropriate adsorption limit with regards to the expulsion of crystal violet dye from fluid arrangements. The harmony adsorption is for all intents and purposes accomplished in 45 min for charcoal and 60 min for rice husk. The amount of crystal violet dye adsorbed (0.4 g of charcoal and 0.5 g of rice husk) increased with an increasing pH and the value of 11 is the best

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

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