# FALL 2017

Math 443 : Partial Differential Equations. A Fourth year course.

Text books :
1. Ian Sneddon , Elements of Partial Differential Equations, McGraw-Hill International Editions (Mathematics Series), 1985
and
2. Richard Haberman :Applied Partial Differential Equations: with Fourier Series and Boundary Value Problems (Fourth Edition), Pearson Education (2004)

Other Books:
R. Dennemeyer : Introduction to Partial Differential Equations and Boundary Value Problems., McGraw-Hill, New York , 1986.

Content of the Course : First Order equations. Method of characteristics., Lagrange-Charpit method, Pfaff systems. Linear Partial Differential Equations with Constant Coefficients.Second order equations, classification, The Method of Seperation of Variable, heat equation,the wave equation, Green's function, Laplace and Poisson's equations.

modelling , Fourier Series

## Course Schedule

• Monday 08.40-10.30 , SA-Z19
• Wednesday 10.40-12.30, SA-Z19
Exams
see previous exams

• (30%) First Midterm Exam pdf file (2005) , (2009) ,(2010), (2011) : November 1
• (30%) Second Midterm Exam pdf file (2005) , (2009) , (2010), (2011), (2017) : December 6
• (40%) Final Exam pdf file (2005) , (2009), (2010), (2011), (2017), Solution(2017) : January 02, 2018,

MAKEUP EXAM: January

Makeup 1 (2009),(2010)
Makeup 2 (2009)
Makeup 2 solutions (2009)

• Solutions of the first midterm exam first midterm ps file (2005) , (2009) , (2010), (2011)
Solutions of the second midterm exam second midterm pdf file (2005) , (2009) , (2010), (2011), (2017)
Solutions of the final exam final exam pdf file (2005) , (2009), (2010), (2011)

Some Exercies: Study problems

## Subjects Covered So far

1. September 18
• Curves and Surfaces in space (R^3)
Chapter 1 (of Sneddon). Sections 1 and 2 including the Problems (pages 1-10)
Homework Set I
2. September 25
• Methods of solutions of dx/P=dy/Q=dz/R
• Applications of the system of equations dx/P=dy/Q=dz/R to dynamical systems in R^3
(this application is not from Sneddon, see the lecture notes).
• Applications of the system of equations dx/P=dy/Q=dz/R to Orthogonal Families of Curves on a surface.
• Pfaffian Differential forms and Pfaffian Differential Equations.
Chapter 1 (of Sneddon). Sections 3,4 and 5 (pages 10-26) including the Problems (pages 15,18, and 26).
3. October 2
• Solutions of Pfaffian Differential Equations in Three Variables
• First Order Partial Differential Equations
Chapter 1 (of Sneddon), Section 6 (pages 26-33) and Problems in page 33, and Miscellaneous Problems in pages 42-43.
Chapter 2 (of Sneddon), Sections 1-4 (pages 44-55) and Problems in page 55.
4. Ocober 9
• First Order Partial Differential Equations
• Integral Surfaces Passing Through a Given Curve
• Cauchy's Method of Characterics.
Chapter 2 (of Sneddon), sections 5,6,8 and Problems in the pages 55,57,59,66.
Homework Set II
5. October 16
• Nonlinear PDEs of First Order
• (Method of Envelopes) Nonlinear PFEs of First Order (section 2.7)
• Partial Differential Equations of Second Order
Chapter 2 Sections (of Sneddon) 7,8,9,10,11. Problems at the end of each section
6. October 23
• Partial Differential Equations with Constant Coefficients
• Partial Differential Equations with Variable Coefficients
Homework Set III
7. October 30
Last week's (6th week) subjects are not included the First midterm.
First Midterm Exam (2005), (2010)
• Partial Differential Equations with Variable Coefficients
• Characteristic Curves of Second Order Equations
8. November 6
• Characteristic Curves of Second Order Equations
• Initial Value Problems, Cauchy Problem .
After this week we shall use the book
"Applied Partial Differential Equations: with Fourier Series and Boundary Value Problems"
by Richard Haberman (Fourth Edition), Pearson Education (2004)

Homework Set IV
9. November 13
• The Heat Equation (Chapter 2)
• The Method of Separation of Variables
• An Initial and Boundary Value Problem With Zero Temperatures at the End Points (2.3)
• Justification of the solution (Convergence problem)
10. November 20
• Solutions of the heat equations with different boundary conditions: Heat Conduction in a Rod With Insulated Ends (2.4.1)
• Heat Conduction in a Thin Circular Ring (2.4.2)
• Laplace's Equation (2.5)
• The Laplace Equation inside a rectangle (2.5.1)
• The Laplace Equation for a circular disk (2.5.2)
11. November 27
• Some properties of Laplace's equation (2.5.4)
• Fourier Series (Chapter 3)
Homework Set V
12. December 4
Second Midterm Exam , (2010)
• The wave equation (Chapter 4)
• Well Posed Problems
• Vibrating String with fixed end Points
13. December 11
• Vibrating String with fixed end Points
• String with infinite length (D'Alembert's solution)
• Regular Sturm-Liouville Problems
14. December 18
• Properties of the Regular Sturm-Liouville Problems (Chapter 5)
• The Rayleigh Quotient
• Boundary Conditions of the third kind
end of the semester

## Course Syllabus

• 1. Sept 18 Chapter 1. Section 1.1. Surfaces in three dimensions Solve exercises at the end of the section on page 7.
• 2. Sept 25 Pfaffian systems and their solutions Chapter 1. Sections 2,3,4,5 completed. Solve all exercises. Sections 7 and 8 will not be done. Solve the Miscellaneous Problems at the end of the Chapter. .
• 3. Feb.21 First order partial differential equations (2.1 - 2.6) Sections 2.1-2.6 completed . You are responsable from all exercies at the end of each Section.
• 4. Mar.14 Nonlinear partial differential equations of first order (2.7-2.11) . Solve the Miscellaneous Problems (M.Ex.1)
• 5. Mar. 21 Second order partial differential equations (3.1-3.5).
• 6. Mar. 28 Characteristic curves and characteristic equations : Solution of the Cauchy problem: Existence and uniquenss of solutions when the data given on (a) Non characteristic and (b) characteristic curves. All sections except section 11 .
• 7. April 8 L aplace's equation (4.1-4.5).
• 8. April 15 Green's function for Laplace's equation All sections of Ch 4. except Section 9. (M. Ex.2)
• 9. April 28 The wave equation (5.1-5.5). Solve all exercises
• 10. May 7. General solutions and Green's function for the wave equation (5.5-5.7). solve all exercises
• 11. May 13 The diffusion (Heat) equation (6.1-6.4).
• 12. May. 21 The use of integral transform and Green's function for the heat equations (6.5--6.6). Solve all exercises at the end of each Sections
13. May.01 Symmetries of partial differential equations.
14. May.08 Group invariant solutions of differential equations.
15. May.15 Review