Lecture 6: 20 September 2010Lecture 6: 20 September 2010 Briefly reviewed key results from previous lecture (inner products, transposes of derivatives, symmetry and negative-definiteness of d2/dx2). One of the things that I've swept under the rug is that up to now, I haven't said anything about whether u(x) is differentiable; if it's not, an inner product like 〈u,u''〉=-〈u',u'〉 might not be defined. If we restrict ourselves to u(x) where 〈u',u'〉 is finite (piecewise differentiable functions), this is called a Sobolev space, and is a subspace of the vector space of all u(x); more generally, one can talk about Sobolev spaces for other differential operators. I'll mostly sweep this under the rug in 18.303, but mainly you should be aware that anyone talking about Sobolev spaces just means they are excluding functions that aren't differentiable enough. Discussed practical implications of the symmetry and negative-definiteness of d2/dx2 (with zero boundary conditions). For the Poisson equation, the lack of a nonzero nullspace means that any solution is unique; if by analogy with matrices we assume that the symmetry means it is diagonalizable in the Hilbert space (a very complicated subject for operators, but more-or-less true for physically realistic symmetric operators, e.g. any square-integrable function has a convergent Fourier series), then we also get existence of the solution for any right-hand side. For the heat/diffusion equation, it means that we have exponentially decaying solutions, and that fast oscillations decay faster (since fast spatial oscillations = large u'' = large eigenvalues). For the wave equation, showed that we can write the solution as a sum of eigenfunctions multiplied by sines and cosines, with the square root of -λ giving the frequency, and two initial conditions giving us the sine/cosine amplitudes. Here, the real λ<0 means that we have oscillating solutions (real frequencies), which are orthogonal (the "normal modes"). As a more general example, considered the Sturm-Liouville operator w(x)-1 [ - d/dx (c(x) d/dx) + p(x) ]. So far, we have considered w=1, p=0, c>0. For pset 2, you consider w>0, c=1, p=0. In class, we consider w=1, p>0, c>0, and show (easily) that it is also real-symmetric positive-definite (with zero boundary conditions), and hence it will have real eigenvalues λ>0, orthogonal eigenfunctions, etcetera. A famous example of such an operator is the Schrodinger equation of quantum mechanics, where c=0 and p(x) is a potential energy. Further reading: For the mathematically inclined, googling "Sobolev space" will turn up lots of more formal definitions; see e.g. section 7.7 of Partial Differential Equations in Action by Salsa for an undergraduate-level (but still rather formal and rigorous) introduction. Much of the theory of these kinds of 1d operators is traditionally called "Sturm-Liouville theory", and can be found under that name in many books (e.g. Methods of Applied Mathematics by Hildebrand, Methods of Mathematical Physics by Courant & Hilbert, and many similar titles). Even Wikipedia has a decent article on the topic under that name.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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