Introduction to Partial Differential Equations

Partial differential equations are the mathematical language used to describe physical phenomena, from heat diffusion to fluid dynamics. Transitioning from ordinary differential equations to partial differential equations can feel daunting without a structured, step-by-step approach. This written course guides you through the foundational theory and practical solving techniques for first- and second-order equations. You will learn how to classify equations, apply classical analytical methods, and understand how these mathematical models represent real-world physical systems. What you will learn: 1. Understand the fundamental classifications of partial differential equations, including elliptic, parabolic, and hyperbolic types. 2. Solve linear first-order equations using the method of characteristics. 3. Apply the separation of variables technique to solve the heat, wave, and Laplace equations. 4. Analyze boundary value problems and understand the role of initial and boundary conditions. 5. Explore modern basic numerical approximation concepts used to solve complex equations computationally. The course begins with core definitions and classification principles before moving systematically into analytical solution methods and physical applications. Each concept is reinforced with detailed written derivations and step-by-step mathematical exercises. This course is designed for beginners in mathematics, physics, and engineering who have a basic background in calculus and ordinary differential equations. Start reading today to build a strong mathematical foundation in partial differential equations.