User:Egm6321.f09/Lecture plan

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"What's the application?" lesson 2

The discoveries were made some 20 years ago in pursuit of a purely scientific problem that seemingly had no

practical relevance. But telomeres have turned out to play a role in two medical areas of vast importance, those of aging and cancer, because of their role in limiting the number of times a cell can divide. 3 Americans Share Nobel for Medicine, NY Times, 5 Oct 2009

Give me a fish, I eat for one day. Teach me to fish, I eat for a lifetime.

Proverb quoted in [Lebesgue integration, S.B. Chae, 1995].

What I tell you three times is true. Lewis Carroll, The Hunting of the Snark, 1874.


All versions[edit]

Fall 2011, Fall 2010, Fall 2009

Recorded lectures, TA user page[edit]

Recorded lectures in E-Learning at UF (password required): Click Continue, type username and password, click at EGM 6321, click Course Content; at "Lecture Videos" link, click at drop down menu, then select "Preview" to go to the web page with lecture video links.

TA user page: Summary of HW statements, for students to interact with TA.

Lecture transparencies, report table[edit]

These Lecture Transparencies were written in real time during the lectures (i.e., not prepared ahead of the lectures). Additional presentations (video, wiki, static html) made in class may not be recorded on these transparencies.

djvu: (install the viewers evince or DjView4)
Mtg 1, Mtg 2, Mtg 3, Mtg 4, Mtg 5, Mtg 6, Mtg 7, Mtg 8, Mtg 9, Mtg 10, Mtg 11, Mtg 12, Mtg 13, Mtg 14, Mtg 15, Mtg 16, Mtg 17, Mtg 18, Mtg 19, Mtg 20, Mtg 21, Mtg 22, Mtg 23, Mtg 24, Mtg 25, Mtgs 26+27 Exam 1, Mtg 28, Mtg 29, Mtg 30, Mtg 31, Mtg 32, Mtg 33, Mtg 34, Mtg 35, Mtg 36, Mtg 37, Mtg 38, Mtg 39, Mtg 40, Mtg 41, Mtgs 42+43 Exam 2, Mtg 44,

Report table

Mourning the Death of Handwriting, By Claire Suddath. Time Magazine, Monday, Aug. 03, 2009.

Op-Art: The Write Stuff, by Inga Dubay and Barbara Getty, NY Times, 8 Sep 2009.



A.C. King, J. Billingham, S.R. Otto, Differential equations: Linear, nonlinear, ordinary, partial, Cambridge University Press, 2003. ISBN-10: 0521016878 ISBN-13: 978-0521016872. UF library 0511078315 (electronic bk.) Google books

D. Zwillinger, Handbook of Differential Equations, Third Edition, Academic Press, 1998. ISBN-10: 0127843965. ISBN-13: 978-0127843964. UF library QA371.Z88 1989, 2 copies, one for in-library use. Google books

O.D. Kellogg, Foundations of potential theory, Dover publications, 1954. UF library QA825 .K4x 1953 Google books

P.M. Morse, H. Feshbach, Methods of theoretical physics, Parts I & II, McGraw-Hill, 1953. UF library QC20 .M6 google books

M. Abramowitz & I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 1972. Read online, Download Wikipedia

N.N. Lebedev, Special Functions & Their Applications, Dover Publications, 1972. ISBN 0486606244 (pbk). UF library QA351.L3613 1972 Google books

K. Oldham, J. Myland, J. Spanier, An atlas of functions, 2nd edition, Springer, 2008. 1st edition, UF library QA331.S685 1987 google books

A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, 1991. UF library QC20.7.O75N5513 1991 Google Amazon

Web references[edit]

Related MIT OpenCourseWare courses

Wolfram|Alpha is ... "learning resource available to your students at no cost that works as a computational knowledge engine. Wolfram|Alpha is not a search engine like Google or Yahoo!, because unlike a traditional search engine, Wolfram|Alpha has the capability to instantly compute the answer to previously unasked questions instead of scouring the web and returning links to pages that already exist. The results are displayed in an easy-to-read, understandable format that can be used as a primary source for educational and academic purposes."

EqWorld, The World of Mathematical Equations. It is a good idea to verify the sources, as the site is not responsible for accuracy and correctness; see Rights and obligations of contributors and website administration.

ODE (Wikipedia): Be careful; always verify the sources.


Vu-Quoc, L., and Olsson, M., ``Formulation of a basic building-block model for interaction of high-speed vehicles on flexible structures, ASME Journal of Applied Mechanics, Vol.56, No.2, pp.451-458, 1989. (pdf)

Vu-Quoc, L., and Olsson, M., ``A computational procedure for interaction of high-speed vehicles on flexible structures without assuming known vehicle nominal motion, Computer Methods in Applied Mechanics and Engineering, Vol.76, pp.207-244, 1989. (pdf)

Vu-Quoc, L., and Tran, V.X., ``Singularity analysis and fracture energy release rate for composites: Piecewise homogeneous-anisotropic materials, Computer Methods in Applied Mechanics and Engineering, John H. Argyris Memorial Issue, Vol.195, No.37-40, pp.5162-5197, 15 July 2006. (pdf)


High-speed maglev trains[edit]

German Transrapid (Maglev), electromagnetic (attraction) maglevs. German Transrapid Emsland 500 km/hr (video, 7:37 min) (500 / 1.6 = 312.5 mi / hr). The Transrapid story (video, history of development, electromagnetic systems vs electrodynamic systems): Part 1 (8:28 min) Part 2 (8:07 min). Shanghai maglev train (video, 5:35 min).

Equations of motion for high-speed vehicles interacting with flexible guideways; see Vu-Quoc & Olsson (1989 a b ): Actually, system of coupled nonlinear 2nd-order ordinary differential equations and partial differential equations.

Japanese Maglev, electrodynamic (repulsion) maglevs, with retractable wheels. Japanese maglev at 581 km/hr (video, 4:43 min) (581 / 1.6 = 363 mi / hr).

French TGV speed record 574.8 km / hr = 357.2 mi / hr (non maglev, wheel-on-rail trains)

Debate on high-speed rail (Sep 2009)[edit]

Siemens Fills Russia’s Need for High-Speed Train, by A.E. Kramer, NY Times, September 24, 2009.

Stimulus Puts High-Speed Rail On The Fast Track, NPR, Morning Edition, 24 Feb 2009; audio 4:10 min.

States Make Pitches For High-Speed-Rail Money, NPR, All Things Considered, 21 Aug 09, audio 7:42 min.

California Edges Ahead In High-Speed-Train 'Race', NPR, All Things Considered, 3 Sep 09.

High-Speed Rail Skeptic Outlines Position, NPR, All Things Considered, 3 Sep 09.

More with Google search for "NPR high speed rail"

Equations of motion[edit]

Coupled nonlinear 2nd-order ODE and PDEs with varying coefficients, which depend on the unknown functions to be solved for. Particularization to linear 2nd-order ODEs with varying coefficients.

Linear 2nd-order ODEs with varying coefficients (L2-ODE-VC)[edit]

Even though this section is about linear 2nd-order ODEs with varying coefficients, many of the methods listed in this section apply to nonlinear ODEs in general, with linear ODEs as particular cases. We present the general nonlinear case first, then particularize to the linear case.





Regular / singular points[edit]

Initial / boundary conditions[edit]

Linear differential operator (1)[edit]

Superposition of solutions[edit]

Applications: Homogeneous L2-ODE-VC[edit]

Helmholtz equation (PDE)[edit]

Helmholtz equation (Wikipedia)


Wave equation[edit]
Vibrating membranes[edit]

Wolfram: Acoustics demos Vibrations of a rectangular membrane

Vibration of circular membrane, animation, Wolfram demo Bessel function Jn Bessel function J2, WolframAlpha

One of Chladni's best-known achievements was inventing a technique to show the various modes of vibration on a mechanical surface.

... Since the 20th century it has become more common to place a loudspeaker driven by an electronic signal generator over or under the plate to achieve a more accurate adjustable frequency. In Ernst Chladni (1756-1827).

Chladni figures Chladni patterns (youtube) Chladni patterns on a vibrating plate excited by an acoustic speaker Vibrating modes of a guitar plate

Acoustics of drums, by T.D. Rossling, Physics Today, Vol.45, No.3, pp.40-47, Mar 1992.

Ansatz (wikipedia): A guessed expression for the solution, as in trial solution.

Helmholtz equation (Wikipedia)

Irrotational, compressible, non-viscous fluids[edit]

Morse & Feshbach (1953), p.307

Sound field animations, D.A. Russell, Kettering University.

Electromagnetic wave propagation[edit]

Separation of variables (PEA2)[edit]

One-dimensional wave equation[edit]

Morse & Feshbach (1953), p.125

Three-dimensional wave equation[edit]

Morse & Feshbach (1953), p.509, p.523

Curvilinear coordinates[edit]
Infinitesimal line (1)[edit]
Laplace operator (1)[edit]
Ellipsoidal coordinates[edit]

Elliptic coordinates (Wikipedia) Ellipsoidal coordinates (Wikipedia) Hyperboloid (Wikipedia)

Morse & Feshbach (1953), p.512

Separated equations: L2-ODC-VC (1)[edit]

Morse & Feshbach (1953), p.523

Reduction of order method 0: Missing dependent variable[edit]

Zwillinger (1998), Sec 55

Nonlinear nth-order ODEs (Nn-ODEs)[edit]

Nonlinear 2nd-order ODEs (N2-ODEs)[edit]

Nonlinear 1st-order ODEs (N1-ODEs) (1)[edit]

Integrating-factor method (1)[edit]

Zwillinger (1998), Sec 79

General nonlinear 1st-order ODEs[edit]
Two exactness conditions[edit]
Special form of N1-ODEs[edit]
Relation for mixed partial derivatives[edit]
Generating exact nonlinear 1st-order ODEs[edit]
Non-exact nonlinear 1st-order ODEs[edit]
Euler integrating factor[edit]
Two particular cases[edit]
General L1-ODE-VC (1)[edit]
A class of exact nonlinear 1st-order ODEs[edit]

L2-ODE-VC with missing dependent variable[edit]

General L1-ODE-VC (2)[edit]

See General L1-ODE-VC (1).

Integrating-factor method: L1-ODE-VC[edit]
PDEs: Not treated here (see PEA2)[edit]

Reduction of order method 1: Exact nonlinear ODEs[edit]

Zwillinger (1998), Sec 63: Applicable to nonlinear ODEs, in particular linear 2nd-order ODEs.

Nonlinear 1st-order ODEs (2)[edit]

See Nonlinear 1st-order ODEs (N1-ODEs) (1) and Integrating-factor method: Nonlinear 1st-order ODEs

Nonlinear 2nd-order ODEs[edit]

General nonlinear 2nd-order ODEs[edit]
Two exactness conditions[edit]
Special form of N2-ODEs[edit]
Relations for mixed partial derivatives[edit]
Generating exact nonlinear 2nd-order ODEs[edit]
Non-exact L2-ODEs with special power form[edit]
Euler integrating factor[edit]
A class of exact L2-ODE-VC[edit]
Legendre L2-ODE-VC (1)[edit]

King et al. (2003), p.31

Nonlinear nth-order ODEs[edit]

Zwillinger (1998), Sec 63, p.289

General Nn-ODEs[edit]

Two exactness conditions[edit]

Special form of Nn-ODEs[edit]
Single exactness relation[edit]
Finding first integral[edit]
Application: N3-ODE[edit]

involving .

Superposition of solution for L2-ODE-VC[edit]

Here, we treat the cases in which at least a solution (homogeneous or particular) can be guessed by inspection; the other solution can then be generated from the guessed solution.

For the cases in which the solutions cannot be guessed by inspection, see Solution by power series: Frobenius method.

Linear differential operator (2)[edit]

Null space[edit]

Homogeneous (complementary) solution[edit]

Euler equations: Special homogeneous Ln-ODE-VC[edit]

Zwillinger (1998), Sec 61

Method of trial solution (undetermined coefficients)[edit]

This method is also known as the method of undertermined coefficients; the terminology "trial solution" is more descriptive since the method involves guessing the solution mathematical expression, called the trial solution, which have unknown coefficients to be determined by substituting the trial solutions into the differential equation.

King et al. (2003), Appendix 5

Zwillinger (1998), Sec 94


Reduction of order method 2: Undetermined factor (1)[edit]

King et al. (2003), p.5

Zwillinger (1998), Sec 85

Undetermined factor[edit]
Homogeneous L1-ODE-VC[edit]
Direct integration[edit]
Integrating factor method[edit]

Particular solution[edit]

Method of trial solution[edit]

Non-homogeneous L2-ODE-VC[edit]

Zwillinger (1998), Sec 94

Non-homogeneous L2-ODE-CC[edit]

Boyce & DiPrima

PDEs: Not treated here (PEA2)[edit]

Variation of parameters[edit]

King et al. (2003), p.7 Zwillinger (1998), Sec 95

Test of linear independence[edit]
The Wronskian[edit]
The Grammian[edit]
Full solution based on homogeneous solutions[edit]

Direct solution with one known homogeneous solution[edit]

Reduction of order method 2: Undetermined factor (2)[edit]

Non-homogeneous L1-ODE-VC[edit]

Integrating factor method[edit]

Solution by power series: Frobenius method (1)[edit]

Zwillinger (1998), Sec 90

Indicial equation, roots[edit]

General rule I: Roots differ by integer[edit]

Special case[edit]

General rule II: Roots differ by non-integer[edit]

General rule III: Roots equal[edit]

Singular points[edit]

18.305 Advanced Analytic Methods in Science and Engineering Fall 2004, Lecture 6 (pdf)

Regular singular points[edit]

18.305 Advanced Analytic Methods in Science and Engineering Fall 2004, Lecture 7 (pdf)

Irregular singular points[edit]

18.305 Advanced Analytic Methods in Science and Engineering Fall 2004, Lecture 8 (pdf)

Legendre functions[edit]

Kellogg (1953), p.125 King et al. (2003), p.31

Motivation: Heat conduction[edit]

Spherical coordinates[edit]


Mathematical physics[edit]

Infinitesimal line (2)[edit]

Laplace operator (2)[edit]

Non-axisymmetric case[edit]
Axisymmetric case[edit]

Separation of variables (2)[edit]

Axisymmetric case[edit]
Separated equations: L2-ODC-VC (2)[edit]
Euler equation (2)[edit]
Legendre differential equation (2)[edit]

Axisymmetric solution[edit]

Method of trial solution[edit]
Legendre polynomials (1)[edit]
Orthogonality (1)[edit]
General expression[edit]
Even-ness, odd-ness[edit]
Boundary condition[edit]
Fourier-Legendre series[edit]
Grammian, linear independence (2)[edit]
Coefficients of series solution[edit]

Lecture transparency p.33-1, Eq.(2) and Eq.(5):

Integration involving transcendental functions: Abramovitz & Stegun, p.77

Circular cylinder (cylindrical) coordinates[edit]

Infinitesimal line (3)[edit]

Laplace operator (3)[edit]

Separation of variables (3)[edit]

Separated equations: L2-ODC-VC (3)[edit]
Bessel differential equation (1)[edit]

King et al. (2003), p.80

Application: Gauss-Legendre quadrature[edit]

Gauss quadrature (wikipedia) Legendre polynomials (wikipedia) Abramovitz & Stegun, p.887 Abramovitz & Stegun, Table 25.4, p.916

Quadrature, cubature[edit]

Numerical integration[edit]

Roots of Legendre polynomials[edit]

Integration points[edit]



Comparison with trapezoidal rule[edit]

"What's the application?" lesson 1[edit]

"What's the application?" lesson 2[edit]

3 Americans Share Nobel for Medicine, By Nicholas Wade, NY Times, Published: October 5, 2009.

The discoveries were made some 20 years ago in pursuit of a purely scientific problem that seemingly had no

practical relevance. But telomeres have turned out to play a role in two medical areas of vast importance, those of aging and cancer, because of their role in limiting the number of times a cell can divide. Wade (2009)

Two homogeneous solutions: Legendre functions[edit]

Legendre polynomials[edit]

Legendre polynomials (wikipedia)

Second homogeneous solutions: Non-polynomials[edit]

Reduction of order method 2 (3)[edit]

Undetermined factor[edit]
Non-polynomial solutions (infinite series)[edit]

General expression[edit]

Even-ness, odd-ness[edit]

General solution of Laplace equation[edit]

Axisymmetric case[edit]

Using the astronomy convention for spherical coordinates, the general solution for the Laplace equation before applying any boundary conditions is:

Historical development[edit]

Application: Attraction of two spheres[edit]

Newtonian potential in 3-D[edit]

Generating function for Legendre polynomials[edit]

Expansion into polynomial series[edit]

Binomial theorem[edit]
Generating function for "n choose r"[edit]
Pochhammer symbol[edit]

Legendre polynomials (1st homogeneous solutions)[edit]

Recurrence relation[edit]

Legendre differential equation[edit]

Solution of Laplace equation[edit]

Orthogonality of Legendre functions[edit]

Application: Laminar flow around a sphere[edit]

Irrotational, incompressible, inviscid flow[edit]

Laplace equation in spherical coordinates[edit]

Fluid flow experiment around a cylinder (video) (not a sphere, but the streamlines are similar to those of a flow around a sphere)

Moving cylinder in a fluid (video)

Non-homogeneous Legendre equation[edit]

Variation of parameters[edit]

Alternative method[edit]

Using only Legendre polynomials[edit]

Bypassing 2nd homogeneous solution[edit]

King et al. (2003), p.44

Legendre equation with non-negative integer order[edit]

Homogeneous equation[edit]

Solution by power series: Frobenius method (2)[edit]

Indicial equation, roots[edit]

General rule I: Roots differ by integer[edit]

Two series solutions[edit]

One finite series[edit]
Legendre polynomials[edit]
One infinite series[edit]

Rodrigues's formula[edit]

Olinde Rodrigues (1795-1851)

Unified general theory of classical orthogonal functions[edit]

Nikiforov et al. (1991), Chap 1.

Fundamental equation of hypergeometric type[edit]

Jacobi equation, functions[edit]

Legendre equation, functions[edit]

Chebyshev equation, functions[edit]

Hermite equation, functions[edit]

Bessel equation, functions[edit]

Transformation by undetermined factor[edit]

Application: Laplacian in cylindrical coordinates[edit]

General Rodrigues's formula[edit]


Boundary value problems[edit]

Asymptotic methods[edit]

Other nonlinear ODEs[edit]

Van der Pol equation[edit]



An asteroid breakup 160 Myr ago as the probable source of the K/T impactor, William F. Bottke, David Vokrouhlický & David Nesvorný, Nature 449, 48-53 (6 September 2007).

Asteroids: Spun in the sun, William F. Bottke, Nature 446, 382-383 (22 March 2007).

Asteroids: How to make a flying saucer, William F. Bottke, Nature 454, 173-174 (10 July 2008).

Asteroid, NASA. Pictures of asteroids Ida, Eros, and Chicxulub (Yucatan) impact, extinction of dinausaurs.

Large asteroid impacting the Earth, simulation

Hubble Space Telescope Captures Rare Jupiter Collision, 07.24.09

Near Earth Objects Program, JPL.

Solar system collision: Set Target Earth (land only), Projectile Rock, Projectile diameter 10 km, Projectile velocity 60 km / sec; click Kaboom; see photo of Chicxulub impact.