Lecture 2: 10 September 2010Lecture 2: 10 September 2010 Started with a very simple vector space V of functions: functions u(x) on [0,L] with u(0)=u(L)=0 (Dirichlet boundary conditions), and with one of the simplest operators: the 1d Laplacian A=d2/dx2. Explained how this describes some simple problems like a stretched string, 1d electrostatic problems, and heat flow between two reservoirs. Inspired by 18.06, we begin by asking what the null space of A is, and we quickly see that it is {0}. Thus, any solution to Au=f must be unique. We then ask what the eigenfunctions are, and quickly see that they are sin(nπx/L) with eigenvalues -(nπ/L)2. If we can expand functions in this basis, then we can treat A as a number, just like in 18.06, and solve lots of problems easily. Such an expansion is precisely a Fourier sine series (see handout). In terms of sine series for f(x), solve Au=f (Poisson's equation) and Au=∂u/∂t with u(x,0)=f(x) (heat equation). In the latter case, we immediately see that the solutions are decaying, and that the high-frequency terms decay faster...eventually, no matter how complicated the initial condition, it will eventually be dominated by the smallest-n nonzero term in the series (usually n=1). Physically, diffusion processes like this smooth out oscillations, and nonuniformities eventually decay away. Using a simple Matlab script to sum the Fourier sine series, animated solutions to the heat equation in a couple of cases, seeing exactly what we expected. As a preview of things to come later, by a simple change to the time-dependence found a solution to the wave equation Au=∂2u/∂t2 from the same sine series, and animated that to see "wavelike" behavior. (This is an instance of what we will later call a "separation of variables" technique.) Further reading: Section 4.1 of the book (Fourier series and solutions to the heat equation).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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