Applied Math   II  G63.2702     Spring 2003

Lecture: 7:10-9:00 pm Thursday, Room 813 WWH.
 Office hours: 5-7pm Thursday or by appointment (, 998-3135).
This course will treat various examples of partial differential equations (PDE's) arising in applications. Examples are chosen
where the intuition provided by the physics can aid in understanding properties of the PDE.
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.
The course will be largely self-contained. Applied Math I (G63.2701) is NOT a prerequisite. No previous PDE course
is assumed. Some knowledge of basic ordinary differential equations and complex variable will be useful.

  Textbooks and Software

The lecture notes will provide the main basis of the course, to be supplemented by reading from various sources.

One text has been  ordered for the course:

E.C. Zachmanoglou and Dale W. Thoe, Introduction to Partial Differential Equations with Applications. Dover Publications Inc.,
New York, 1976 (1986).

Some material will come from the book used in Applied Math I, Principles of Applied mathematics, by Keener (Perseus Books, 2000). This text will be available on reserve in the CIMS library.

Other books which have been placed on reserve are:

Fritz John,  Partial Differential Equations
Garabedian, Partial Differential Equations
R.B. Guenther and John W.  Lee , Partial Differential Equations of Mathematical Physics and Integral Equations
G.B. Whitham, Linear and Nonlinear Waves

This list will be updated as required. All of these books are recommended should you want to purchase
a supplementary text, but they vary in emphawsis and mathematical level.

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

Tentative syllabus:
The course will deal first with wave propagation and the wave equation, including applications in gas dynamics, traffic flow,
and water waves. The second segment will deal with the heat equations and problems involving diffusion. Applications to
heat transfer and developmental biology will be considered. Harmonic functions and potential theory will be considered,
with applications to electrostatics and fluid flow. The final section of the course will take up special applications,
depending upon the interests of the class.

Various mathematical techniques will be developed as required, including generalized functions and Fourier series and

Lecture 1 January 23.

Introduction to applied mathematics of partial differential equations. Derivation of basic equations in mathematical physics. Linear and nonlinear problems Gas dynamics and the acoustic approximation, leading to the linear wave equation.  The heat equation. Harmonic functions. Well-posed and ill-posed problems for the equations.  (Good and bad equations.) Bad solutions of good equations, and  uniqueness of  the solution of the initial value problem for the heat equation.

Reading: Text, Chapter VI.

Lecture 2 January 30.

An example of two different PDE representations for the same model: Eulerian and Lagrangian gas dynamics. The
elastic string as a realization of the one-dimensional wave propagation.

Analysis of the  linear wave equation in one dimension. Basic solutions techniques for the initial-value (Cauchy) problem (IVP).
D'Alembert's solution. Concepts of range of influence and domain of dependence.
Initial-boundary-value problem (IBVP) by reflection. The bursting of a spherical balloon. Uniqueness of solution of
the IVP and IBVP. Introduction to generalized functions.

Reading: Text 291-304. Homework 1, to be handed in February 6: Download pdf file.
Download AdobeReader.

Lecture 3 February 6.

Application of generalized functions to construction of Riemann function for the 1-D wave equation.
Examples of IBVP and solution by separation of variables. Kirchoff's solution of the IVP in three dimensions
The Riemann function in three dimensions. Huygen's principle.

Reading: Text 271-290, 308-315. Homework 2, to be handed in February 13: Download pdf file.

Lecture 4 February 13.

Exploding balloon as an example of Kirchoff's solution. Download handout  pdf file.
The method of decent and solutions of  the wave equation in two dimensions. Download handout pdf file.
Integral curves and surfaces of a vector field. Definition of linear and quasilinear
PDEs of first order. General methods for solution. Characteristic curves and their physical realization.
Examples. A quasilinear wave equation solved along characteristics.

Reading: Text 24-33, 64-71. Guenther and Lee 19-34.  Homework 3, to be handed in February 20
(Note homework 3 was modified Feb. 13, in case you downloaded an earlier version): Download pdf file.

Lecture 5 February 20.

The quasilinear wave equation u_t+uu_x=0. Expansion fans. Discontinuous solutions and the formation of shocks
in the quasilinear equation u_t+F(u)_x=0. Calculation of shock speed and trajectory. A model for traffic flow.
Traffic flow versus car path. The red light problem.

Reading: Text 72-81, Whitham 68-72. Homework 4, to be handed in Feb. 27: Download pdf file.

Lecture 6 February 27.

The computation of shock path (continued). Light versus heavy traffic.
The problem of a light turning red in light traffic, then green. Multilane roads.

Linear dispersive waves. Superposition and the Fourier representation.
Phase versus group velocity. The behavior of dispersive systems for long times
and the method of stationary phase.

 Reading: Whitham, chapter 11. Homework 5, due March 6 : Download pdf file.

Lecture 7 March 6.

Stationary phase continued. Wave energy and group velocity.

Elements of the  Fourier integral. The heat equation in  one dimension: Probabilistic meaning.
The fundamental solution , by similarity and by Fourier transform. Poisson's solution of the IVP.
Boundary-value problems. Duhamel's  principle.

 Reading: (For lectures 7 and 8)  Text chapter IX . Homework 6, due March 13: Download pdf file.

Lecture 8 March 13.

Reaction-diffusion models of pattern formation in biology. 

Lecture 9 March 27.

The maximum principle for the heat equation. Uniqueness issues.
Solution of the IBVP by reflection. The fundamental solution in higher
dimensions.  Convection versus conduction of heat. The advection-diffusion problem
in Lagrangian coordinates.

An application of the diffusion equation: lichen growth.

Reading: pp. 343-350 of text, pp. 174-76 and 179-80 of John.
Homework 7, due April 3. Download pdf file.

Lecture 10 April 3.

Laplace's equation and potential theory.  Problems arising in physics.
Funbdamental solution in 2 and 3 dimensions. Relation in 2D to analytic functions
of a complex variable. Multipole solutions in 3D. Boundary-value problems.  Uniquess
of solutions of Dirchlet and Neumann problems.
Poisson's integral in 2D. Representation theorem in 3D. The mean value property.
Green's functions.

Reading: From chap. VII of text:  sects. 1,2,4,5,9,10,13.  Homework 8, due April 10. Dounload pdf file.
NOTE:This is the last homework to be handed in for students working on a term project.

Lecture 11 April 10.

Water waves as an example of dispersive waves.
General linear theory for deep and shallow water. Nonlinear shallow-water theory. Derivation of the Kortewed de Vries equation.
The solitary wave. Fully nonlinear waves in shallow water. Charteristics and Riemann invariants for this
quasi-linear system.

Lecture 12 April 17.

Other aspects of quasi-linear hyperbolic systems of PDE's. Characteristics and invariants, simple wave systems.
Applications to shallow water and to gas dynamics in one dimension.

Reading: pp. 380-390 of text.

Lecture 13 April 24.

Perturbation methods.  Orders and asymptotic expansions.  Asymptotic sequences and expansions. Regular and singular perturbations.  Inner and outer expansions and matching,
using a simple model boundary layer problem.  The Prandtl boundary-layer theory.

Lecture 14 May 1

Perturbation  methods continued

NOTE: All outstanding problem sets should be turned in by May 5.  Projects due May 12.