This dissertation presents a numerical investigation of the two-dimensional incompressible Navier-Stokes equations, focusing on the classic Lid-Driven Cavity problem. The study develops a computational fluid dynamics (CFD) solver from first principles using the Finite Difference Method (FDM) on a structured Cartesian grid, providing a funda- mental understanding of the pressure-velocity coupling in viscous flows. The numerical framework employs the Projec- tion Method, originally proposed by Chorin, to enforce the incompressibility constraint. This operator-splitting technique solves an intermediate velocity field which is subsequently projected onto a divergence-free space via a Pressure Poisson Equation (PPE). The governing equations are discretized using second-order central differences for spatial derivatives and a first-order explicit Euler scheme for time integration. The solver is validated at a Reynolds number of Re = 100. The simulation results successfully capture the characteristic flow features, including the primary central vortex and the corner recirculation eddies. Quantitative validation is performed by comparing the vertical centerline velocity profiles against the established benchmark data of Ghia et al. The results demonstrate excellent agreement with the benchmark solutions, confirming that the developed solver correctly resolves the physics of wall-bounded shear flows. This work establishes a robust foundational framework for simulating viscous incompressible flows and highlights the efficacy of the Projection Method for fundamental CFD applications.

The rapid rise of Large Language Models (LLMs) has shifted the battleground of digital misinformation. Unlike human-written fake news, machine-generated disinformation often employs subtle linguistic patterns that evade conventional detection systems. Although Deep Learning models can effectively identify synthetic text, they frequently operate as "black boxes," failing to offer the transparency needed for sensitive real-world applications. To address this, we introduce a hybrid architecture that merges the contextual strengths of DistilBERT with the sequential analysis capabilities of Bidirectional Long Short-Term Memory (BiLSTM) networks. Crucially, we incorporate SHapley Additive exPlanations (SHAP) to decode the model's decision-making process, visualizing exactly which words or tokens tip the scales toward a specific classification. Tests on the benchmark Fake or Real News dataset [1], supplemented by a 5-fold cross-validation protocol to ensure robust statistical validation, show our framework achieves an average accuracy of 96.92% ± 0.18%. By leveraging Explainable AI (XAI), we confirm that the model identifies actual semantic anomalies rather than merely overfitting to background noise, offering a more trustworthy foundation for automated fact-checking systems.