Mathematical modeling and numerical simulation constitute the twin pillars of modern computational science, providing quantitative frameworks for understanding, predicting, and controlling complex dynamic systems across virtually every domain of science and engineering. This article reviews the foundational principles of mathematical modeling—encompassing ordinary, partial, and stochastic differential equations—alongside the principal numerical simulation strategies employed to solve them, including Runge–Kutta time integration, finite difference methods (FDM), finite element methods (FEM), and adaptive mesh refinement. Benchmark evaluations conducted on canonical dynamic systems—the Lorenz attractor, epidemiological SIR model, structural vibration system, and climate sub-system model—demonstrate that solver selection critically governs prediction accuracy, numerical stability, and computational efficiency. Results reveal relative errors spanning 0.34 % to 1.20 % and stability indices between 0.88 and 0.97 depending on the system class and integration method. The article further discusses emerging hybrid approaches that couple physics-based differential equations with machine learning surrogates, identifying their potential to dramatically reduce computational cost while preserving physical interpretability. Proper formulation of boundary and initial conditions, rigorous error analysis, and systematic model validation are identified as non-negotiable prerequisites for reliable simulation outcomes in applied settings.