# SPRING 2019

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.

## Course Schedule

• Tuesday 10:30-12:20, SA-Z02
• Thursday 15:30-17:20, SA-Z02
Exams
see previous exams

• (25%) First Midterm Exam pdf file (2005) , (2009) ,(2010), (2011), (2019) : November 6
• (25%) Second Midterm Exam pdf file (2005) , (2009) , (2010), (2011), (2017), (2019), : December 11
• (25%) Final Exam pdf file (2005) , (2009), (2010), (2011), (2017), Solution(2017), (2019) : November 6 : January 12, 2020,(09:00-11:00, SA-Z19)
• (25%) Homework

MAKEUP EXAM:

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