Lecture 3: 13 September 2010Lecture 3: 13 September 2010 Began with a warning: functions are more complicated than vectors, and it is possible to come up with crazy counter-example functions where 18.06-like techniques don't work, if you try. Gave examples of a function lacking a sine series and a function where the sine series obviously does not converge to the right result. Much of a rigorous pure-math course, like a course on functional analysis, has to do with carefully circumscribing the set of functions and operators that will be considered, in order to exclude the pathological cases that don't resemble finite vectors and matrices. Now, we will go back to the happy land of finite-ness for a while, by learning to approximate a PDE by a matrix. This will not only give us a way to compute things we cannot solve by hand, but it will also give us a different perspective on certain properties of the solutions that may make certain abstract concepts of the PDE clearer. We begin with one of the simplest numerical methods: we replace the continuous space by a grid, the function by the values on a grid, and derivatives by differences on the grid. This is called a finite-difference method. Went over the basic concepts and accuracy of approximating derivatives by differences; see handout. Armed with center differences (see handout), went about approximating the 1d Laplacian operator d2/dx2 by a matrix, resulting in a famous tridiagonal matrix known as a discrete Laplacian. The properties of this matrix will mirror many properties of the underlying PDE, but in a more familiar context. Further reading: chapter 1 of the book, especially section 1.2 on finite differences.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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