Aim of this course is to provide a comprehensive introduction to the theory and applications of hyperbolic partial differential equations (PDEs), a fundamental class of equations that describe wave propagation, conservation laws, and various physical phenomena. The course begins by introducing the basic concepts and characteristics of hyperbolic PDEs, including well-posedness, existence, uniqueness, and stability of solutions. We will explore both linear and nonlinear systems, focusing on canonical examples such as the wave equation, transport equations, and systems of conservation laws.

Key topics include the method of characteristics, Riemann invariants, shock waves, and rarefaction waves. The course will also delve into numerical methods for solving hyperbolic systems, discussing finite difference schemes, and addressing the challenges of capturing discontinuities and shocks. Applications in fluid dynamics, traffic flow, and other areas will be examined to highlight the relevance and utility of hyperbolic PDEs in modeling real-world problems.