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MIT 18 303 - Lecture Notes

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Lecture 1: 8 September 2010Lecture 1: 8 September 2010 General overview of what a PDE is and why they are important. Discussed examples of some typical and important PDEs (see handout, page 1). With non-constant coefficients (the most common case in real-world physics and engineering), even the simplest PDEs are rarely solvable by hand; even with constant coefficients, only a relative handful of cases are solvable, usually high-symmetry cases (spheres, cylinders, etc.) solvable. Therefore, although we will solve a few simple cases by hand in 18.303, the emphasis will instead be on two things: learning to think about PDEs by recognizing how their structure relates to concepts from finite-dimensional linear algebra (matrices), and learning to approximate PDEs by actual matrices in order to solve them on computers. Went through 2nd page of handout, comparing a number of concepts in finite-dimensional linear algebra (ala 18.06) with linear PDEs (18.303). The things in the "18.06" column of the handout should already be familiar to you (although you may need to review a bit if it's been a while since you took 18.06)—this is the kind of thing I care about from 18.06 for this course, not how good you are at Gaussian elimination or solving 2×2 eigenproblems by hand. The things in the "18.303" column are perhaps unfamiliar to you, and some of the relationships may not be clear at all: what is the dot product of two functions, or the transpose of a derivative, or the inverse of a derivative operator? Unraveling and elucidating these relationships will occupy a large part of this course. Note: Just as in 18.06, for as long as possible I will stick with real numbers and real functions, but at some point complex numbers will become hard to avoid. When that happens, we will have to switch from transposes AT to conjugate-transposes A*.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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