| 539 DIARY as of | 12/04/04 | ||
| Class Date | Class # | HW | Topics explored |
| due | |||
| Tue, Aug 31, 04 | 1 | Scaling, nondimensionalization. Flagpole. | |
| Thu, Sep 2, 04 | 2 | In Lab: intro to Maple | |
| Tue, Sep 7, 04 | 3 | Exact soln of flagpole, resonances. Holmes 1.3 Big-O, little-o, definitions and examples. | |
| Thu, Sep 9, 04 | 4 | Holmes 1.4 Asymptotic sequences and expansions: tie in with iterative expansion in 1/J in flagpole. Start Calc of Var. | |
| Tue, Sep 14, 04 | 5 | #1 | Minimal surf of rev example. Derived Euler-Lagrange eqn, and its 1st integral in case f doesn't depend on x. Solved shortest path, brachistocrone. |
| Tue, Sep 21, 04 | 6 | #2 | Solution of minimal surface problem: multiple stationary points. Use of Maple fsolve() and evalf(Int()). Extensions: (i) integral depends on multiple functions, shortest path in 3d; (ii) integral constraint, Lagrange multiplier, isoperimetric problem. |
| Thu, Sep 23, 04 | 7 | Extensions, cont'd: (iii) free and other BCs, (iv) higher-order problems, (v) multiple independent variables, (vi) non-integral constraints. Hamilton's principle, the Lagrangian, Lagrange's eqns of motion. Started compound pendulum. | |
| Tue, Sep 28, 04 | 8 | #3 | Completion of Lagrangian for compound pendulum, formation & linearization of EL eqns, small amplitude solution. Use of Maple diff, subs, linalg[eigenvals,eigenvects], functions, and @ for function composition. |
| Thu, Sep 30, 04 | 9 | EXAM#1 | |
| Tue, Oct 5, 04 | 10 | General solution of double pendulum, quasiperiodicity, comparison with experimental (paint-stick) observations. Variational formulation of elastic hanging chain. | |
| Thu, Oct 7, 04 | 11 | Elastic chain solution. Singular inextensibility limit. Inextensible uniform chain solution. Optimal variable cross-section mooring chain problem. | |
| Tue, Oct 12, 04 | 12 | Optimal chain problem completed. Dynamics of moored ship. Implicit function: slope of solution locus of system of constraints. | |
| Thu, Oct 14, 04 | 13 | #4 | Group solution of mooring spring constant problem. (HW#5 involves Maple fsolve() and linalg[linsolve(),nullspace(), cols(),delcols()]. ) |
| Tue, Oct 19, 04 | 14 | Mention of table rocking problem. Integral equations: 3 examples in applications. Fredholm & Volterra eqns. Solution of Volterra eqn of convolution type by Laplace transform. Iterative solution method applied to ODE IVP (Picard iteration). | |
| Thu, Oct 21, 04 | 15 | #5 | Integral equations, cont'd: Neumann iteration. Fredholm equations with separable kernels. |
| Tue, Oct 26, 04 | 16 | Finished and reviewed Fredholm alternatives for separable kernels. Self-adjoint operators, real symmetric kernels. | |
| Thu, Oct 28, 04 | 17 | #6 | Solution by eigenfunction expansion. Example (Logan 1.6): eigendata by conversion to ODE BVP. Shortcoming of eigenfunction expansion (slow nonuniform convergence). Numerical solution of IE: splines, collocation. |
| Tue, Nov 2, 04 | 18 | EXAM#2 | |
| Thu, Nov 4, 04 | 19 | Exam review, integral equation equivalent to delay differential equation (DDE). Some simple DDEs and solutions, instability in damped harmonic oscillator with delayed restoring force. Overview of heat. | |
| Tue, Nov 9, 04 | 20 | Overview of heat transfer. | |
| Thu, Nov 11, 04 | 21 | A similarity solution as a fundamental solution for heat equation, and from it, solution for temporally distributed heating, initial temperature specification, integral as solution (erf) of semi-infinite slab with sudden-onset heating. | |
| Tue, Nov 16, 04 | 22 | #7 | Solution for semi-infinite body with Neumann and Dirichlet BCs: images. Method of images for finite slab: practical for short times only. F&M 7.6: Boundary integral methods. Newtonian surface heating of semiinfinite body: an integral equation for the surface temperature. |
| Thu, Nov 18, 04 | 23 | Approximate and exact solution of IE for surface temperature: by Neumann iteration. | |
| Tue, Nov 23, 04 | 24 | Asymptotic expansions for integrals, Laplace integrals, Laplace's method. Gamma function, Watson's Lemma, integration by parts. Erfc example, non-convergence of asyptotic series. | |
| Tue, Nov 30, 04 | 25 | #8 | Review of heat problems solved in F&M Ch 7. Discussion of sidewalk solar heating problem. F&M Ch 8. Separation of variables, Fourier coefficients. Adjoint of 2nd order linear differential operator. Orthogonality of eigenfunctions of self-adjoint op. |
| Thu, Dec 2, 04 | 26 | Fourier solution of Newtonian heating of finite slab: analysis of convergence properties. Steady state for sinusoidal surface heating (1/8 cycle lag). | |
| Tue, Dec 7, 04 | 27 | #9 | REVIEW |
| Thu, Dec 9, 04 | 28 | EXAM#3 |