Lecture 7: 22 September 2010 More examples: Laplacian in d dimensions: showed that with Dirichlet/Neumann boundary conditions (at least over any finite domain Ω), this operator is real-symmetric and negative definite/semidefinite, respectively. Key to the proof is the divergence theorem, which generalizes "integration by parts" to multiple dimensions. Similarly for c .non-constant coefficient operator x for any positive function c(x)>0. This immediately tells us (if we take diagonalizability somewhat on faith) about existence and uniqueness of solutions to Poisson's equation, that diffusion in any medium has exponentially decaying solutions, etcetera, that waves are oscillating, just as in 1d. Considered general Dirichlet boundary conditions (u = any function b(x) on the boundary dΩ), and showed (similar to pset 1) that we can turn this back into zero Dirichlet boundaries by writing u=u0+b and modifying the right-hand side accordingly. Discussed Neumann boundary conditions. For the diffusion equation, these are often the u is aright choice. By considering conservation of mass (integral of u), showed that mass flow per unit area per unit time. In a domain where nothing is flowing in or out (e.g. diffusion of salt in a closed container), this means that u is zero in the normal direction at the boundary. By the same token, this gives conservation of mass in the diffusion equation. More generally, derived a conservation law for any problem of the form Au=∂u/∂t: if m(x) is a function in the left nullspace of A, then it is easy to show that the quantity 〈m,u〉 is a constant, hence conserved. In the case of the diffusion equation with Neumann boundary conditions, the left nullspace (= right nullspace) is just constant functions, and this means that the integral of u(x) is conserved. Further reading: The "integration by parts" that we did with the Laplacian is sometimes referred to as Green's first identity (although it really is just integration by parts via the divergence theorem); see also section 1.6 of Partial Differential Equations in Action bySalsa. Positive-definite (or sometimes semidefinite) operators like - are sometimes called elliptic operators.MIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit:
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