Advanced Partial Differential Equations: Theory and Applications

Partial differential equations are the mathematical language used to describe the physical laws of our universe, from fluid dynamics to quantum mechanics. For researchers, engineers, and applied mathematicians, a deep theoretical and practical understanding of these equations is essential. This text-based course guides you from foundational definitions to advanced analytical techniques, helping you solve complex physical problems with mathematical rigor. You will build a strong conceptual framework, starting with essential terminology before progressing to advanced solution methods and modern numerical approximations. What you will learn: Understand the classification, boundary conditions, and physical origins of second-order partial differential equations; Apply energy methods and maximum principles to prove uniqueness and stability of solutions; Master Sobolev spaces and weak formulations to analyze equations that lack classical solutions; Explore modern numerical methods, including finite element analysis and basic computational solver frameworks; Practice solving elliptic, parabolic, and hyperbolic equations through structured, step-by-step mathematical derivations. This course begins with a thorough review of fundamental concepts and classical solution techniques, then transitions into advanced weak formulations and modern computational frameworks. It is designed for students, researchers, and engineers who have a basic background in calculus and ordinary differential equations, requiring no prior advanced analysis experience. Start mastering the mathematical tools needed to analyze and solve complex physical systems today.