Linear Partial Differential Equations: Analysis and Numerics with Julia

Partial differential equations govern everything from heat flow to wave propagation, yet bridging the gap between pure mathematics and practical computer simulation can feel daunting. This course provides a clear, step-by-step pathway to understanding and solving linear partial differential equations (PDEs). You will learn to analyze fundamental equations—such as the heat, wave, and Poisson equations—through the intuitive lens of linear algebra, while simultaneously building modern, efficient numerical solvers using the Julia programming language. What you'll learn: - Understand the core concepts of linear PDEs, including boundary conditions and classifications. - Analyze the heat, wave, and Poisson equations using linear algebra analogies and matrix structures. - Implement finite-difference methods in Julia to discretize and solve continuous differential equations. - Explore finite-element techniques to transform complex physical problems into solvable matrix equations. - Practice writing clean, modern Julia code to simulate physical systems and verify analytical solutions. - Apply stability and convergence analysis to ensure your numerical models are accurate and reliable. You will start with foundational definitions, key terminology, and basic physical concepts before exploring classical analytical solutions. From there, you will transition into numerical methods, translating mathematical theory into functional Julia programs through detailed written explanations and code exercises. This course is designed for beginning students, engineers, and scientists looking to build a strong foundation in PDEs and scientific computing, requiring only a basic familiarity with calculus and linear algebra. Start mastering the mathematics and computation of physical systems today.