Lecture 19: PDE I: Examples/theory, derivative operators (part I)

The professor announces the due dates for homework assignments, mentions an optional homework assignment focusing on ODE integrators, and provides a brief overview of the remaining class schedule, including the final exam location and date. CS205a class 119: Homework 7 due, optional HW8 (ODE integrator review) available. Final exam is Dec 12th in Gates B3 (not the usual classroom!). Last lecture (PDEs part 2) is Wednesday; final review is Friday. Course reviews are open; help with creating course notes offered for independent study units. Lecture covers PDEs, focusing on theoretical aspects and discretization. Examples include Navier-Stokes (unsolved problem!), Maxwell's equations, and Laplace's equation for interpolation. The lecture revisits ordinary differential equations (ODEs), focusing on initial value problems and their application in physics, particularly Newton's second law. The professor uses the example of simulating springs to illustrate the concept. The instructor details the final exam schedule, location (emphasizing it's not in the usual classroom), and offers additional office hours to help students prepare. The professor also encourages students to utilize office hours for help and mentions the course review system. This segment introduces partial differential equations (PDEs), highlighting their complexity compared to ODEs. The professor discusses applications in various fields like fluid dynamics and image processing, emphasizing the lack of guaranteed solutions for some PDEs and the importance of verifying solvability before proceeding with solutions. This segment delves into the Navier-Stokes equations, a complex set of equations used to simulate fluid dynamics. The speaker highlights the equations' non-linearity and the significant challenge they pose to mathematicians, even explaining the million-dollar prize offered for proving their solvability in three dimensions. The discussion underscores the complexities and open questions within the field of fluid simulation. This segment introduces Maxwell's equations, describing their role in electromagnetism, and transitions to Laplace's equation, a simpler PDE relevant to the course's homework. The speaker connects Laplace's equation to interpolation problems, setting the stage for a discussion on its practical applications in computer graphics and data interpolation. This segment focuses on the application of PDEs to interpolation problems, particularly on irregular domains. The speaker explains the challenge of interpolating functions on complex shapes and introduces the concept of using least squares methods and energy minimization to solve this problem. The connection to previous course material on least squares is highlighted, emphasizing the practical application of theoretical concepts.