Syllabus Detail

Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 118B: Partial Differential Equations

Approved: 2003-03-01, Spitzer & Shkoller
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
“Introduction to Partial Differential Equations, 1st Edition” by Walter Strauss. ($50.00)
Search by ISBN on Amazon: 471548685
Suggested Schedule:

Lecture(s)

Sections

Comments/Topics

1

6.1-6.4

Second order Linear Differential equations in 2D and 3D.

Harmonic Functions

Laplace equation; examples, Maximum Principle, Uniqueness of solutions, and Symmetries. Extension to elliptic differential equations.

Laplace equation on Rectangles and Cubes.

Poisson’s Formula, Mean Value Property, and Maximum Principle.

Laplace equation on Circles, Wedges, and Annuli.

2

7.1-7.4

Green’s Identities and Green’s Functions

Green’s First Identity, Maximum Principle, Dirichlet’s Principle.

Green’s Second Identity.

Green’s Functions in general.

Green’s Function for Half-Space and Sphere.

3

9.1-9.4

Diffusion and Wave Equation in unrestricted 2D and 3D.

Energy and Causality.

Kirchhoff’s Formula in 3D, and the solution in 2D.

Inhomogeneous wave equation in 3D.

Diffusion equation in 2D and 3D.

4

10.1-10.3

Boundary Problems in 2D and 3D.

Separation of Variables, revisited.

Vibrations of a Drumhead, Bressel functions (see also 10.5)

Sketch of 3D Wave Equation in a ball.

5

11.1 & 11.6

General Eigenvalue Problems: An Introduction

Minimum Principle

Asymptotics of Eigenvalue

Additional Notes:
Recommendation: use matlab, mathematica or maple for demonstration in class early on, and frequently.
Learning Goals:
AS A CAPSTONE: In the second and third courses of the MAT 118 sequence, students undertake an in depth study of advanced methods in partial differential equations. These methods include integral operators, spectral decomposition, energy methods, calculus of variations, and many other advanced techniques. Applications of these techniques to standard and specialized equations arising from mathematics and physics will be demonstrated. Students will gain mastery over technical aspects of the field, and develop their ability to communicate at a capstone level, commensurate with that expected of an undergraduate degree in mathematics.