King Fahd University of Petroleum and Minerals

Department of Mathematics & Statistics

Syllabus Math 470

Semester I, 2009-2010 (091)

Instructor: Dr. Faisal Abdul-Karim Fairag

Course:             Math 470 (Partial Differential Equations)

Text Book:         Beginning Partial Differential Equation. by P. O’Neil.

Reference: Applied partial Differential Equations by John Ockendon

Objectives:         Firstorder quasilinear equations. Lagrange method and Characteristics. Classificationof linear second order PDEs. Brief review of separation of variables. The onedimensional wave equation: its solution and characteristics. Cauchy problem forthe wave equation. Laplace's equation: The maximum principle, uniquenesstheorems. Green's function. Neumann's function. The heat equation in onedimension.

Week

Sec

## Topic

Suggested

Homework

1

1.1

Introduction, Linear first order PDE’s, Characteristics

2, 4, 7, 8 p3-4

2

1.2

1.3

1.4

Quasilinear first order equations, the Cauchy problem, Characteristic method, General solutions

10, 11, 12p11

1, 3, 5 p15-16

1, 3, 5, 7, 9, p22

3

2.1

2.2

2.3

2.4

Second order PDE’s in two variables: Classification by characteristics, Canonical forms, Second order Cauchy problem

1, 3, 5, 7, 9, page25

2(a, c), 3, p29

2, 3, p33

1, 3, 5, 6, p36

7, 9, p37

4

2.5

4.1

3.4

4.8

The wave equation in 1D: The Cauchy problem and d’Alembert’s formula,

Review of  Fourier series and sine and cosine expansion

The wave equation: Fourier Series Solutions, Separation of variables.

1 p45

3, 5, 9, 10, 13, p116-117

1, 3, 7,9  p87-88

1, 5, 11, 14,15a  p158

# "A+"   Monday 26 Oct, 2009: Exam I,  6:00 PM  "A+"

5

4.3

6.1

6.2

6.4

The Characteristic  Triangle  Domain of dependence and range of influence, Well-Posedness.

Dirichlet and Neumann ( setting the Problem)

Some Harmonic Functions

Maximum Principle & Mean value property

1, 2 p130

170 p266

8 p246

3, 4, 5, 6, p250

4, 6 p261-262

6

6.5

3.6

6.9

8.2

Existence Uniqueness and well-posed

The Fourier Transform

Dirichlet Problem Upper Half-plane (Fourier Transform)

Numerical Approximation of  Solutions (Finite Difference)

1  p266

3, 12, 15, 22 p101

7

3.5

6.9

6.10

The Fourier Integral

Dirichlet Problem Upper Half-plane (Fourier Integral)

Dirichlet Problem Upper the Right Quarter-Plane

1, 6 p93

1, 4, p279

1 p282

# J Eid Al-Adha Vacation: 19 Nov - 4 Dec, 2009  J

8

6.7

6.12

6.13

Dirichlet Problem for a disk

The Neumann Problem

The Neumann Problem for a Rectangle

4, 6 p271

9

6.15

6.14

The Neumann Problem for the upper half-plane

The Neumann Problem for a disk

"A+"   Saturday, 12 Dec 2009: Exam II,   6:00PM  "A+"

10

5.1

5.2

The Cauchy Problem and Initial Condition

The Weak Max. Principle

11

5.3

5.4

Solutions on Bounded Intervals

The Heat Equation on real line

12

5.5

5.7

The Heat equation on the Half-line

The Nonhomogeneous Heat Equations

"A+"   Monday 4 Jan, 2009 : Exam III,   6:00PM   "A+"

13

8.2

Numerical Approximation of  Solutions(Finite Element)

Matlab and PDE toolbox

14

6.16

7.2

7.3

Green's Function for a Dirichlet Problem

A Hilbert Space Approach

Distributions and an Existence Theorem +Lax Milgram Th.

15

6.17

Conformal Mapping Technique

Review and catch-up

1. The date of the final exam will be announced by the Registrar. The Final Exam is comprehensive.