Taiye Oyedepo

This study presents the development, analysis and numerical performance of a new numerical algorithm designed for solving Volterra integro-differential equations of the second kind. The method is derived using a power series approach with interpolation and collocation techniques, leading to a continuous and discrete block formulation suitable for solving linear and nonlinear Volterra integro-differential equations of the second kind. The fundamental properties of the algorithm such as order, error constant, consistency, zero-stability, convergence and absolute stability are rigorously examined and the method is shown to satisfy all necessary conditions for reliability and convergence. To assess its accuracy and efficiency, three numerical examples involving linear and nonlinear Volterra integro-differential equations with known exact solutions are solved. The results demonstrate that the developed algorithm consistently produces extremely small absolute errors, significantly outperforming several existing numerical methods. The algorithm proves to be highly accurate, stable and suitable for a wide range of volterra integro-differential problems.

The implementation of block algorithm for the direct solution of third order oscillatory problems of the form has been studied in this paper. The methodology is carried out using the procedure of interpolation and collocation. The analysis of the paper was analyzed. Also, on the applications, the results obtained shows that is highly efficient and can handle different equations for which the algorithm was designed for. This is because the computed solution matches the exact solution. In fact, the method obviously performed better than the ones with which we compared our results with. This shows that, the algorithm generates results very fast. Thus, there is economy of time in the computation.

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.

The Human Development Index (HDI) is a measure that evaluates achievements in human development based on fundamental components of quality of life which influence individual productivity levels. This study aims to analyze the impact of Population Size, Percentage of Poor Population, Open Unemployment Rate (TPT), and Gini Ratio on the Human Development Index (HDI) in the Bangka Belitung Islands Province from 2018 to 2023. The panel data regression method was employed, combining time series and cross-sectional data to provide more accurate information. The findings reveal that the HDI of the Bangka Belitung Islands Province has increased year-on-year, though it remains below the national average. The selected model was the Fixed Effect Model (FEM) based on the results of the Chow and Hausman tests. Analysis indicates that Population Size has a positive and significant effect on HDI, while the Percentage of Poor Population has a negative and insignificant effect. Additionally, the Open Unemployment Rate (TPT) has a positive and significant effect on HDI, whereas the Gini Ratio has a negative and significant effect. Collectively, the four independent variables contributed 95.44% to HDI. These findings are expected to inform government efforts to improve human development quality, such as through poverty alleviation and highlight the need for careful population management to balance employment opportunities in the region. Furthermore, the government should work to reduce the unemployment rate through policies that focus on job creation, increasing workforce competitiveness, and addressing inequality, as reflected by the Gini ratio. This can be achieved by expanding equitable access to education and healthcare.

This study demonstrates the derivation of a two-step block scheme simulation through a linear block approach. The scheme's fundamental properties were thoroughly analyzed and found to fulfill all necessary conditions. The research focused on examining specific classes of oscillatory differential equations and comparing them to established methods. The findings indicate that the newly proposed methods exhibit superior accuracy and faster convergence compared to the existing methods investigated in this research. Consequently, the results highlight the improved precision and quicker convergence achieved with the new method. All computations were executed using Maple 18 software

This work presents a collocation computational algorithm for solving linear Integro-Differential Equations (IDEs) of the Fredholm and Volterra types. The proposed method utilizes shifted Legendre polynomials and breaks down the problem into a series of linear algebraic equations. The matrix inversion technique is then employed to solve these equations. To validate the effectiveness of the suggested approach, the authors examined three numerical examples. The results obtained from the proposed method were compared with those reported in the existing literature. The findings demonstrate that the proposed algorithm is not only accurate but also efficient in solving linear IDEs. In order to present the results, the study employs tables and figures. These graphical representations aid in displaying the numerical outcomes obtained from the algorithm. All calculations were performed using Maple 18 software.

In this study, the Volterra-FredholmIntegro-Differential Equations (VFIDEs) are solved computationally using first-kind Chebyshev polynomials as basis functions. The technique converts VFIDEs into a system of unknown Chebyshev coefficients linear algebraic equations, which are then solved via matrix inversion. Five numerical examples were solved in order to assess the scheme's correctness and effectiveness. The findings reveal that the approach outperforms the comparison results in the literature

This work focuses on fractional calculus, which is calculus with fractional derivatives. The ideal is that we have the first derivative, which is velocity, and the second derivative, which is acceleration, and that we can have any derivative between the first and second derivatives. To this end, the Homotopy Perturbation Technique (HPT) is used to approximate the solution of Fractional Integro-Differential Equations (FIDEs) with the Caputo derivative, which provides less rigorous works with improved accuracy. To demonstrate the method, some numerical examples are provided. The findings achieved by the current method are found to be comparable to the exact result.