18.303 Midterm Topic Summary November 1, 2010 • Eigenfunctions and eigenvalues: given the eigenfunctions/eigenvalues ofˆAu f, ˆA, using them to solve equations such as ˆ= Au = ∂u/∂t, and ˆAu = ∂u2/∂t2 (or other variations). – Important special solvable cases: Fourier series and Bessel functions. Separation of variables in space (and time, but we view separation in time as decomposition into eigenfunctions). Inner products �u, v�, adjoints, definiteness, and properties of Aˆ. Given • Aˆand an inner product, how to find Aˆ∗ and how to show whether Aˆis self-adjoint (Hermitian or real-symmetric), positive/negative definite or semidefinite, and what the consequences of these facts are for eigenvalues and eigenfunctions or for equations in terms of Aˆlike ˆAu = ∂u/∂t and so on. – e.g. why the heat/diffusion equation has exponentially decaying so-lutions (and why oscillations are damped out especially rapidly). – understand how to integrate by parts with �. • Null spaces of operators (left and right) and their consequences on solv-ability, uniqueness and (in the case of left nullspaces) conservation laws for ˆAu = ∂u/∂t (e.g. conservation of mass for difusion equation). • Effect of boundary conditions on definiteness, self-adjointness, nullspaces. Turning general Dirichlet and Neumann boundaries into zero Dirichlet and Neumann boundary conditions (i.e. boundary conditions as modifying the right-hand-side). • The Rayleigh quotient and the min–max (variational) theorem, and its consequences. Guessing the form of the smallest-|λ| solutions using the min–max theorem. Finite-difference discretizations: • – Analyzing the order of accuracy of a given discretization (with Taylor expansions). 1– Writing the discretized A to have the same self-adjoint/definite prop-erties as Aˆ(e.g. by writing A in terms of DT D. – Effect of boundary conditions. – Finite-difference discretizations in more than 1 dimension. • Green’s functions and inverse operators. – Relationship between ˆand Green’s function ˆx, � = x −A−1 AG(� x�) δ(��x�). (And how properties of Aˆeffect properties of G, e.g. self-adjointness gives reciprocity.) – Solving G in simple cases (using the delta function, not using ugly limits), especially empty space Ω = Rd . – How solutions Aˆ−1f are made from the Green’s function. – How the Green’s function G0 of empty space relates to Green’s func-tions in inhomogeneous systems or systems with boundaries. Born– Dyson series and Born approximations. Delta functions and distributions. • – Definition, regular vs. singular distributions. Differences from ordi-nary functions. – Distributional derivatives. – Solving PDEs with δ on the right-hand side (e.g. finding Green’s functions, above). 2MIT 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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