On a nonlocal boundary value problem for the equation fourth-order in partial derivatives

В данной статье изучается нелокальная задача для уравнения четвертого порядка в которой доказывается существование и единственность решения этой задачи. Решение построено явно в виде ряда Фурье, обоснованы абсолютная и равномерная сходимость полученного ряда и возможность почленного дифференцирования решения по всем переменным. Установлен критерий однозначной разрешимости поставленной краевой задачи. In this article, we study a nonlocal problem for a fourth-order equation in which the existence and uniqueness of a solution to this problem is proved. The solution is constructed explicitly in the form of a Fourier series; the absolute and uniform convergence of the obtained series and the possibility of term-by-term differentiation of the solution with respect to all variables are substantiated. A criterion for the unique solvability of the stated boundary value problem is established.

Photodegradation and kinetics of edible oil refinery wastewater using titanium dioxide

Edible oil refinery wastewater (EORW) is one source of environmental pollution in Nigeria. The treatment of EORW before discharge into the environment remains a significant challenge in the edible oil refinery industries. This research was aimed at photocatalytic treatment of EORW using a batch photocatalytic reactor with titanium dioxide photocatalyst. We investigated the physicochemical parameters: chemical oxygen demand (COD), biological oxygen demand (BOD5), oil and grease, phenol, chloride (Cl-), total suspended solids, sulfate (SO42-), and phosphate (PO43-) using American Public Health Association methods. The results showed that the reduction efficiency of the treated EORW with TiO2 catalyst ranged between 65.8% (PO43-) and 87.0% (COD), and the improvement in efficiency was 54.1% (pH) and 60.8% dissolved oxygen. However, the results showed no significant difference (p<0.05) in the control treatment without catalyst. The biodegradability of EORW increased from 0.196 to 0.32. It was observed that the optimum values were an initial EORW concentration of 100 mL/L, irradiation time of 90min, catalyst dose of 1.25 g/L, and an agitation speed of 900 rpm. The kinetics of the photodegradation process was well described by the pseudo-first-order equation (R2>0.96) and pseudo-second-order equation (R2>0.98). The intra-particle diffusion model fairly represented the diffusion mechanism with an R2 value of 0.806. The treated EORW met the most acceptable water quality standards for discharged effluent according to the maximum permissible limits of the Nigerian National Environmental Standards and Regulations Enforcement Agency.

Relativistic Quantum Motion of an Electron in Spinning Cosmic String Spacetime in the Presence of Uniform Magnetic Field and Aharonov-Bohm Potential

In this manuscript, we study the relativistic quantum mechanics of an electron in external fields in the spinning cosmic string spacetime. We obtain the Dirac equation and write the first- and second-order equations from it, and then, we solve these equations for bound states. We show that there are bound state solutions for the first-order equation Dirac. For the second-order equation, we obtain the corresponding wave functions, which depend on the Kummer functions. Then, we determine the energies of the particle. We examine the behavior of the energies as a function of the physical parameters of the model, such as rotation, curvature, magnetic field, Aharonov-Bohm flux, and quantum numbers. We find that, depending on the values of these parameters, there are energy nonpermissible levels.

Gaussian exponential regression method for modeling open circuit voltage of lithium-ion battery as a function of state of charge

Purpose The purpose of the article is to develop an equation to accurately represent OCV as a function of SoC with reduced computational burden. Dependency of open circuit voltage (OCV) on state of charge (SoC) is often represented by either a look-up table or an equation developed by regression analysis. The accuracy is increased by either a larger data set for the look-up table or using a higher order equation for the regression analysis. Both of them increase the memory requirement in the controller. In this paper, Gaussian exponential regression methodology is proposed to represent OCV and SoC relationships accurately, with reduced memory requirement. Design/methodology/approach Incremental OCV test under constant temperature provides a data set of OCV and SoC. This data set is subjected to polynomial, Gaussian and the proposed Gaussian exponential equations. The unknown coefficients of these equations are obtained by least residual algorithm and differential evolution–based fitting algorithms for charging, discharging and average OCV. Findings Root mean square error (RMSE) of the proposed equation for differential evolution–based fitting technique is 35% less than second-order Gaussian and 74% less than a fifth-order polynomial equation for average OCV with a 16.66% reduction in number of coefficients, thereby reducing memory requirement. Originality/value The knee structure in the OCV and SoC relationship is accurately represented by Gaussian first-order equation, and the exponential equation can accurately describe the linear relation. Therefore, this paper proposes a Gaussian exponential equation that accurately represents the OCV as a function of SoC. The results obtained from the proposed regression methodology are compared with the polynomial and Gaussian regression in terms of RMSE, mean average, variance and number of coefficients.

A Review of Gum Hydrocolloid Polyelectrolyte Complexes (PEC) for Biomedical Applications: Their Properties and Drug Delivery Studies

The utilization of natural gum polysaccharides as the vehicle for drug delivery systems and other biomedical applications has increased in recent decades. Their biocompatibility, biodegradability, and price are much cheaper than other materials. It is also renewable and available in massive amounts, which are the main reasons for its use in pharmaceutical applications. Gum can be easily functionalized with other natural polymers to enhance their applications. Various aspects of the utilization of natural gums in the forms of polyelectrolyte complexes (PECs) for drug delivery systems are discussed in this review. The application of different mathematical models were used to represent the drug release mechanisms from PECs; these models include a zero-order equation, first-order equation, Higuchi, simplified Higuchi, Korsmeyer–Peppas, and Peppas–Sahlin.

Mathematical Modeling and Functional Order Pursuits with Separated Dynamics

This work is devoted to the study of the pursuit problem in controlled systems described by a fractional-order equation with divided dynamics. For fixed player controls, representations of solutions are established in the form of analogs of the Cauchy formula using generalized matrix functions. Sufficient conditions are obtained for the possibility of completing the pursuit. Specific types of fractional differential equations and models of fractional dynamical systems are considered. The qualitative dynamics, issues of stability and controllability of such systems are discussed. Considered, try which, the motion of the equation is described with irrational orders. Problems of the type under study are encountered in modeling the processes of economic growth and in problems of stabilizing dynamic systems.