Lecture 19: 22 October 2010Lecture 19: 22 October 2010 So far, we have discussed three types of problems: 1. Au=f where A is self-adjoint and positive (or negative) definite. This is sometimes called an elliptic problem. 2. Au=∂u/∂t where A is self-adjoint and negative definite (hence exponentially decaying solutions). This is sometimes called a parabolic problem. 3. Au=∂u2/∂t2 where A is self-adjoint and negative definite (hence oscillating solutions). This is sometimes called a hyperbolic problem. In order for us to learn more about the latter two problems, we need to get more serious about time dependence. In particular, we need to learn to simulate time-dependent problems, leading to the subject of (finite-difference) time-domain numerical methods. Gave a brief argument that this brings in a new problem to study: in addition to discretizing the time derivative, we also need to worry about stability. More on this in the next lecture.MIT OpenCourseWarehttp://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and NumericsFall 2010 For information about citing these materials or our Terms of Use, visit:
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