Applied Math   II  G63.2702     Spring 2001
Lecture: 7:10-9:00 pm Tuesday, Room 613 WWH.
 Office hours: 5-7pm Tuesday or by appointment (childres@cims.nyu.edu, 998-3135).
This course will treat various examples ot theory and application of partial differential equations (PDEs).  Both linear and nonlinear problems  will be considered. Specific applications will be selected based upon
 the interests of the class. Homework will be assigned, collected, and graded, and there will be a final examination.
Numerical work using the MATLAB program will be included as desired.
 

  Textbooks and Software

One text has been  ordered for the course:

 Guenther, Ronald B., and Lee, John W. Partial Differential Equations of  Mathematical Physics and Integral Equations, Dover Publications, Inc., New York, 1988.

Other books will be placed on reserve in the Courant Library, as required.

MATLAB is accessible from UNIX and ACF accounts. There will be a MATLAB tutorial for students unfamiliar with MATLAB.

Lecture 1 January 16.

Introduction to applied mathematics of partial differential equations. Wave equation governing an elastic string and acoustics. Heat conduction and the heat equation. Potential theory and harmonic functions. The telegrapher's equation.

Reading: G&L chapter 1.

Lecture 2 January 23.

The homogeneous wave equation in one dimension. Characteristics. The initial-value problem (IVP) and D'Alembert's solution. Range of influence and domain of dependence. Examples of solutions of the IVP.  The initial-boundary-value problem (IBVP). The balloon burst as a one-dimensional IBVP. The  IBVP by separation of variables.

Reading: G&L sections 4.1,4.2, 4.3 to page 103. Homework 1, to be handed in January 30: Download TeX file. Download pdf file.
Download AdobeReader.

Lecture 3 January 30.

Uniqueness by the energy method. The inhomogeneous wave equation.
Use of generalized functions to find the fundamental solution to the wave equation in one
dimension.  The fundamental solutions in two and three dimensions. Kirchoff's solution of the IVP in three dimensions.
The balloon burst reconsidered.

Reading: G&L sections 4.3, 10.4, 10.5. Homework 2, to be handed in February 6: Download TeX file. Download pdf file.

Lecture 4 February 6.

Quasilinear first-order equations and the method of characteristics. A basic nonlinear wave equation, simple waves,  expansions fans. In the context of a model of traffic flow:  acceleration from a red  light, characteristics, fans.
Shocks and the determination of shock velocity .

Reading: G&L  sections 2.1, 12.3. Homework 3, to be handed in February 20: Download TeX file. Download pdf file.


NOTE: The February 13 class is canceled. It will be replaced by a class on May 1, same room and time.

SPECIAL HOMEWORK: These are problems to develop ideas outlined at the end of lecture 3. Due February 20
with set 3. Download TeX file. Download pdf file.

 

Lecture 5 February 20.

Review of simple waves, fans, shocks, with examples. Calculation of traffic flow through a series of red lights. The critical density for an efficient light.  Overcrowded traffic. Computing vehicle path. Contact discontinuities and shocks.

Reading: G&L section 12.3 through p. 504, Whitham, Linear and Nonlinear Waves, pp. 68-72. On reserve in the CIMS library.

The critical density in the red-green light problem for light traffic is revealed! Click here

Lecture 6 February 27.

Dispersive waves. Concept of phase velocity. Wave superposition.
Asymptotic evaluation of wave form using the method of stationary phase.
Application to linear theory of surface waves over deep water. Effect of a bottom.

Reading: Whitham pp. 363-374. Homework 4, to be handed in march 6: Download TeX file. Download pdf file.

Lecture 7 March 6.

Group velocity and the propagation of wave energy.
Nonlinear wave systems. Characteristics and Riemann invariants. Gas dynamics and shallow-water theory. Application
to the breaking of a dam.

Reading: G&L section 12.2, Whitham 374-379, 454-458.

Lecture 8 March 20.

Diffusion and heat conduction. Basic equation with Fick's law. Diffusion of probability.
The fundamental solution by similarity. Uniqueness issues.
The IVP for  the infinite interval.  The Duhamel integral. Fourier methods for the IBVP.  Elements of the Foureir integral.
Fourier derivation of the fundamental solution.
Reading: G& L 352,353,  section 9.2,  166-172.  Kevorkian, pp. 5-27.   Homework 5, to be handed in March 27: Download TeX file, Download pdf file.


 

Lecture 9 March 27.

General IBVP for 1D heat equation. Use of reflection. The Greens function.
The maximum principle and its application to prove uniqueness of solutions
of the heat equaion.

Homework 6, to be handed in April 3: Download TeX file, Download pdf file.


Lecture 10 April 3.

Potential theory. BVP's for  Laplace's equation. Uniqueness of solutions.
Separation of variables and the Poisson formula in 2D.  Fundamental solutions.  Integral identities
and Green's functions.  Applications in electrostatics  (capacitance) and fluid dynamics (potential flow).
Reading: G&L  8.1-8.3.


Lecture 11 April 10.  Single and double-layer potentials. The maximum principle for harmonic functions.
Exterior problems in spherical polar coordinates:  Legendre's equation,
special functions, potential flow past a body and electrical capacity of a sphere.

Reading: G&L 8.4,8.6 , Kevorkian, Partial Differential Equations  (on reserve),  section 2.12.


Lecture 12 April 17. Application 1: Stability and bifurcation of solutions in nonlinear problems.

Lecture 13 April 24.  Application 2: Turings's theory of diffusive morphogenesis

Lecture 14 May 1 (Make-up). Application 3: PDE's in theories of flying and swimming.

Projects due May 4.