Lecture 5: 17 September 2010Lecture 5: 17 September 2010 Reviewed inner products of functions. A vector space with an inner product (plus a technical criterion called "completeness" that is always satisfied in practice) is called a Hilbert space. Note that we include only functions with finite norm 〈u,u〉 in the Hilbert space, which throws out a lot of divergent functions and means that everything has a convergent Fourier series. As usual, we have to ignore finite discrepancies at isolated points, or otherwise you can have 〈u,u〉=0 for u(x) nonzero (there is a rigorous way to do this, which we will come to later). Defined the transpose AT of a linear operator (or adjoint A*, for complex vector spaces): whatever we have to do to move it from one side of the inner product to the other. For matrices and ordinary vector dot products, this is equivalent to the "swap rows and columns" definition. For differential operators, it corresponds to integration by parts, and depends on the boundary conditions as well as on the operator and on the inner product. Showed that with u(0)=u(L)=0 boundary conditions, (d/dx)T=-(d/dx)...very closely analogous to what we found with the finite-difference derivative matrix D. Furthermore, (d2/dx2)T is real-symmetric (also called "Hermitian" or "self-adjoint"). Showed that the proof of real eigenvalues from 18.06 carries over without modification; similarly for the proof of orthogonal eigenvectors, hence the orthogonality of the Fourier sine series. (Diagonalizability is more tricky, but in practice physical real-symmetric operators are almost always diagonalizable; the precise conditions for this lead to the "spectral theorem" of functional analysis.) Not only that, but showed that d2/dx2 is negative-definite on this space, since 〈u,u''〉=-∫|u'|2, and u'=0 only if u=constant=0 with these boundary conditions. So, many of the key properties of d2/dx2 follow "by inspection" once you learn how to transpose operators (integrate by parts). Not only that, but you can obtain the same properties for many operators that cannot be solved analytically. For example, showed that the operator d/dx [c(x) d/dx], which is the 1d Laplacian operator for a non-uniform "medium", is also real-symmetric positive definite if c(x)>0, given the same u(0)=u(L)=0 boundary conditions. Finally, gave a quick Matlab demo of the finite-difference equivalent of d/dx [c(x) d/dx], and showed that it indeed is real-symmetric negative-definite, with real, negative eigenvalues and orthogonal eigenfunctions, and the eigenfunctions are vaguely like sin(nπx/L) except that they are distorted when c(x) is not constant. Further reading: Textbook, section 3.1: transpose of a derivative. The general topic of linear algebra for functions leads to a subject called functional analysis;a rigorous introduction to functional analysis can be found in, for example, the book Basic Classes of Linear Operators by Gohberg et al.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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