Matthew Olanrewaju Oluwayemi

This study obtained the energy levels and examined the partition function (Z) of a quantum system described by an Eckart-like potential model. By adopting the Greene–Aldrich approximation scheme for the centrifugal term, the radial Schrödinger equation (SE) is solved and the analytic expression of the energy eigenvalues is obtained. The ro-vibrational Z is computed by explicitly incorporating the rotational quantum number, a feature often neglected or misapplied in many studies. This result is used to evaluate the key thermodynamic properties (TP), including the Gibbs free energy (G), entropy (S), and enthalpy (H). Numerical analysis reveals that the Z increases monotonically with temperature, while the G decreases in accordance with statistical thermodynamics. The S exhibits saturation-like behaviour at higher temperatures, while the H displays convex growth with increasing thermal energy. Parametric studies demonstrate that the Eckart-like potential allows for the controlled tuning of TP, with variations in the potential parameters, including the screening parameter, having distinct effects. The results generalise existing models, reproduce the Hulthén potential under specific conditions, show the effect of the rotational quantum number of TP, and provide new insights into the ro-vibrational statistical mechanics of exponential-type potentials.

Integro-differential equations (IDEs) seamlessly blend differential and integral operators, offering a versatile mathematical framework for modelling various physical, biological, and engineering systems. Analytical solutions to these equations are often unattainable due to their complexity, necessitating efficient numerical approaches. This study introduces the Legendre Collocation Method (LCM), leveraging shifted Legendre polynomials to approximate solutions for Volterra and Fredholm IDEs. By transforming the IDEs into a system of linear equations, the proposed method achieves high accuracy and computational efficiency. Numerical experiments on representative IDEs validate the approach, showing excellent agreement with exact solutions and outperforming existing methods in terms of absolute error. These results underscore the potential of the LCM in solving complex IDEs with broad applicability across disciplines.

A major interest in geometric function theory is to provide insight into the behaviour of analytic functions. In this study therefore, the authors defined a subclass of univalent functions by means of a generalised multiplier transformation and obtained some geometric properties related to the bounds of the coefficient for a certain class of analytic function. The results obtained in the study extend some existing results in literature.

Various researchers have considered different forms of bi-univalent functions in recent times, and this has continued to gain more attention in Geometric Function Theory (GFT), but not much study has been conducted in the area of application of the certain probability concept in geometric functions. In this manuscript, our motivation is the application of analytic and bi-univalent functions. In particular, the researchers examine bi-univalency of a generalized distribution series related to Bell numbers as a family of Caratheodory functions. Some coefficients of the class of the function are obtained. The results are new as far work on bi-univalency is concerned.

In this study, the numerical solution of the Volterra-integro differential equations was obtained by applying the variational iteration strategy with the shifted Vieta-Lucas polynomials. The proposed method builds the shifted Vieta-Lucas polynomials for the Volterra-integro differential equation which are then used as basis functions for the approximation. Numerical examples were given to establish the effectiveness and dependability of the recommended approach. Calculations were performed using Maple 2022 software.