Introduction to Partial Differential Equations

Partial differential equations (PDEs) are the mathematical foundation used to describe physical phenomena ranging from heat flow and wave propagation to financial markets. Understanding how to formulate and solve these equations is essential for anyone entering the fields of applied mathematics, physics, or engineering. This course guides you from basic definitions to the core analytical techniques used to solve classical first-order and second-order PDEs. You will transition from a basic understanding of calculus to confidently classifying, setting up, and solving fundamental partial differential equations. Through structured text-based lessons and practical mathematical exercises, you will develop a strong analytical intuition for how physical systems are modeled mathematically. What you'll learn: - Understand foundational concepts, definitions, and the classification of partial differential equations - Solve first-order linear and quasi-linear PDEs using the method of characteristics - Master classical second-order equations including the wave, heat, and Laplace equations - Apply boundary and initial conditions to find unique physical solutions - Practice analytical techniques such as separation of variables and Fourier series - Explore modern computational contexts and qualitative properties of PDE solutions This course begins with essential terminology and mathematical prerequisites, ensuring you build a solid theoretical foundation. You will then progress systematically through first-order equations, classification methods, and classical boundary value problems. This course is designed for beginners in advanced mathematics, engineering students, and physics enthusiasts who have a basic background in single-variable calculus and ordinary differential equations. No prior knowledge of partial differential equations is required. Begin your journey into advanced mathematical modeling and master the foundations of partial differential equations today.