In this course, 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.
Note 1: Introduction to IVPs / The Elementary Theory of IVPs
Note 2: The Elementary Theory of IVPs (continued)
Note 3: Euler's Method
Note 4: High-Order Taylor Methods
Note 5: Runge-Kutta Methods
Note 6: Runge-Kutta Methods (continued) / Error Control and the Runge-Kutta-Fehlberg Method
Note 7: Multistep Methods
Note 8: High-Order Equations and Systems of Differential Equations
Note 9: Stability
Note 10: Midterm Review
Note 11: Stiff Differential Equations
Note 12: The Linear Shooting Method
Note 13: The Shooting Method for Nonlinear Problems
Note 14: Finite-Difference Methods for Linear Problems
Note 15: Finite-Difference Methods for Nonlinear Problems
Note 16: The Rayleigh-Ritz Method
Note 17: Elliptic Partial Differential Equations
Note 18: Parabolic Partial Differential Equations
Note 19: Hyperbolic Partial Differential Equations
Note 20: An Introduction to the Finite Element Method