MATH 107: NUMERICAL DIFF EQNS

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.