In this class, we study numerical methods for ordinary differential equations (ODEs) and partial differential equations (PDEs). Some ODEs are solved with proper initial conditions, and this kind of problems is called an initial-value problem (IVP). We start to learn fundamental numerical methods to approximate the exact solution for the IVP with a first-order ODE, such as Euler’s methods, Taylor methods, Runge-Kutta methods, and Multistep methods. We also study advanced techniques to effectively reduce errors occurring in numerical methods (Adaptive algorithm and Extrapolation methods). Moreover, when such numerical methods are applied, we deal with theoretical issues that mainly include stability, consistency, and convergence. Another kind of ODE problems is a boundary-value problem (BVP). Its solution is approximated by Shooting methods used with the numerical methods for IVPs. Finite-difference methods (FDMs) are useful methods to get approximate solutions for both ODEs and PDEs. Other sophisticated methods, Rayleigh-Ritz and Finite-element (FE) methods, are introduced.
Zoom Meeting Information
- Discussions (Tue, Thu 1:00 – 2:00 pm)
- Office hours (Mon, Wed 10:30 – 11:30 am)
Zoom Meeting Videos
- Week 1: Discussion 1 (Mar. 31), Discussion 2 (Apr. 2)
- Week 2: Discussion 3 (Apr. 7), Discussion 4 (Apr. 9)
- Week 3: Discussion 5 (Apr. 14), Discussion 6 (Apr. 16)
- Week 4: Discussion 7 (Apr. 21), Discussion 8 (Apr. 23)
- Week 5: Discussion 9 (Apr. 28), Discussion 10 (Apr. 30)
- Week 6: Discussion 11 (May 5), Discussion 12 (May 7)
- Week 7: Discussion 13 (May 12), Discussion 14 (May 14)
- Week 8: Discussion 15 (May 19), Discussion 16 (May 21)
- Week 9: Discussion 17 (May 26), Discussion 18 (May 28)
- Week 10: Discussion 19 (Jun. 2)
Discussion Notes
- Note 1_Mar. 31: Introduction to IVPs, The Elementary Theory of IVPs
- Note 2_Apr. 2: The Elementary Theory of IVPs (continued)
- Note 3_Apr. 7: Euler’s Method
- Note 4_Apr. 9: High-Order Taylor Methods
- Note 5_Apr. 14: Runge-Kutta Methods
- Note 6_Apr. 16: Runge-Kutta Methods (continued) / Error Control and the Runge-Kutta-Fehlberg Method
- Note 7_Apr. 21: Multistep Methods
- Note 8_Apr. 23: Multistep Methods (continued) / High-Order Equations and Systems of Differential Equations
- Note 9_Apr. 28: High-Order Equations and Systems of Differential Equations (continued) / Stability
- Note 10_Apr. 30: Stiff Differential Equations
- Note 11_May 5: The Linear Shooting Method / The Shooting Method for Nonlinear Problems
- Note 12_May 7: The Shooting Method for Nonlinear Problems (continued)
- Note 13_May 12: Finite-Difference Methods for Linear Problems
- Note 14_May 14: Finite-Difference Methods for Nonlinear Problems
- Note 15_May 19: The Rayleigh-Ritz Method
- Note 16_May 21: Elliptic Partial Differential Equations
- Note 17_May 26: Parabolic Partial Differential Equations
- Note 18_May 28: Hyperbolic Partial Differential Equations
- Note 19_Jun. 2: An Introduction to the Finite-Element Method