African Journal of Mathematics (Pure Science) | 02 February 2005
Nonlinear Differential Equations Framework for Power-Grid Forecasting in Uganda Using Finite-Element Discretization with Error Bounds Analysis
K, i, z, z, a, M, u, k, a, s, a
Abstract
Nonlinear differential equations are essential in modelling complex systems such as power grids, which exhibit nonlinearity due to interactions between various components like generators and loads. Finite-element methods are employed to discretize the nonlinear differential equations governing the power-grid dynamics, enabling numerical simulations that account for spatial variations and discontinuities. Error analysis is conducted to assess the reliability of these forecasts under finite-element approximation. Theoretical analysis and finite-element simulation results demonstrate the feasibility of applying nonlinear differential equations for accurate power-grid forecasting in Uganda’s complex grid environment. The framework provides a robust method for improving grid stability and reliability predictions. Recommend further empirical validation through real-world data integration, alongside exploring potential improvements to the error bounds analysis technique. The analytical core is $\hat{y}<em>t=\mathcal{F}(x</em>t;\theta)$ with $\hat{\theta}=argmin_{\theta}L(\theta)$, and convergence is established under standard smoothness conditions.