Math 445 Home

Supplementary notes,

Mathematica notebooks.

Homework problems and solutions

Computer software:

Many software packages have now built-in routines for solving differential equations. They include:

Mathematica is multipurpose software, that allows symbolic, numeric and graphic manipulation of data. Among its many built-in packages, there are powerful Differential equation solvers (analytic and numeric). 

Another choice is MatLab. It has among other two handy specialized differential equations packages designed by J. Polking for graphic solutions of ODEs and systems.

DE Modeling (ch.1, 13, 14)

  1. Basic PDE models, problems, methods: ppt.
  2. Balance-conservation summary
  3. Basic fluid dynamics and acoustics (short review) (13.2) .
  4. Mechanical systems, Elasticity and Maxwell equations of electrodynamics: pdf (13.1)
  5. Supplementary topics (notes on balance laws and variational principles).
  6. Classical variational problems (Euler-Lagrange equations, Hamiltonian formalism, examples) tif (14.3). Short version
  7. More fluids (tif; pdf). Elastic beam: tif, nb
  8. Soving DE 's with Mathematica

Method of characteristics (1.2; 14.1)

New DEtools package for Mathematica 6 by Selwyn Hollins (follow installation instructions)

Sample.nb

1.    Linear 1-st order equations: pdf, Characteristics.nb

2. Quasi-linear equations and shocks. Additional notes (PDF) and shock.nb

3. Classification of 2nd order PDEs, initial and boundarty conditions; wellposedness (notes)

Waves in 1D (2.1; 2.2; 3.1; 3.2; 13.3)

  1. Characteristic coordinates.
  2. Notes on 1D waves (d'Alembert solution, reflection and scattering). Fundamental solutions of hyperbolic equations in free space
  3. Mathematica nb: d’Alembert, Hammer blow , Multiple reflections, Scattering via NDSolve.nb

Diffusion (Ch. 2-3, 12)

  1. Diffusion (2.3-2.5, 3.3; additional notes pdf) notebook.
  2.   Fundamental solutions of heat and wave problems on R.
  3. Multiple reflection for heat and waves: nb (ch.3)
  4. Stationary and periodic point sources : Green's functions of the Laplace ploblem ( "flux condition" tif )  
  5. Half-line-diffusion and Boundary sources ( nb),
  6. Transform methods: Fourier transform (nb) (12.3-4); Laplace transform (12.5). Periodic BC and Poisson summation
  7. Conclusions: heat and wave equations

Method of Separation ; Eigenvalue Problem: Sturm-Liouville and Fourier Series: (ch. 4; 5)

  1. Separation and Sturm-Liouville problem (ch.4; pdf; nb This nb uses a special package RootSearch, that you can place in the folder ...\Mathematica\6.0\AddasOns\ExtraPackages\).
  2. Review of fourier-series (ch.5).
  3. Eigen-function expansion method: ODE, Elliptic BVP . Notebooks: oscillator, Heat-diffusion with delta-source.
  4. Green's functions: i) eigen-expansion; ii) additional notes; iii) expanded version (via Fourier method)
  5. Completeness and orthogonality
  6. Other examples: Dynamic BVP (string-mass) and Flexible chain (tif; mass-string-nb; heavy-chain-nb). Elastic beam.

Harmonic functions (Ch.6). Green's functions and potential theory (Ch.7)

  1. Basic properties, connection to analytic functions, Poisson formula (pdf)
  2. Eigen-function expansion method for special regions (tif). Stationary heat distribution in annulus: nb
  3. Green's functions for Laplacian and Potential theory. Reflection and half-space problems: pdf
  4. Coordinate change in Lapalcian, Inversion and Conformal mapping: tif, pdf.
  5. 2D point sources: nb
  6.  Applications:
    1. Stationary and moving boundary heat source (nb)
    2. Fluid flow passed obstacles: (i) cylinder and plate pdf; (ii) more detailed fluids pdf ; (iii) nb
    3. * Rayleigh-Ritz approximation: tif; nb;
    4. * Double and single layer potential (Fredholm integral method): pdf
    5. * Green's functions for related parabolic and hyperbolic problems: tif
  7. Additional notes: Green's functions and Fourier transform

Multi-D eigenmodes:(ch.10 - 11:3,4,5)

  1. Rectangular and disk modes: Demo nb. Primer on Bessel functions pdf ; nb. Eigenexpansion method: nb

  2. Sherical harmonics: nb , and spherical density plot (demo) nb

  3. Applications of Laplace eigenfunctions. Boiling egg (heat propagation in cylidrical, spherical regions): tif, nb

  4. Summary disk-ball eigenmodes pdf. Summary of 1D and multi-D eigenvalue problems: pdf.

Applications of eigenvalue problem in linear and nonlinear models: pdf

  1. Applications to nonlinear systems: (i) Linearized stability and bifurcations; (ii) Feedack control: thermostat
  2. Summary: separation and Eigenfunction expansion method tif
  3. Harmonic functions and sources in disk and ball, stationary and moving sources, vibrations tif.
  1. Bifurcations. Reaction-diffusion model: tif; nb

3.    Rayleigh-Benard convection: pdf, tif. Lorenz-Saltzman system (bifurcations and chaos): nb. Animations (mov): 3D chaotic Lorenz and 110D Rayleigh-Benard

Basic Fourier transform: tif

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