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MATH 443 - PARTIAL DIFFERENTIAL EQUATIONS

(1999-2000, Spring 2002, 2004-2005) Fall 2009-2010

Math 443 : Partial Differential Equations. A Third year course.
Text book : Ian Sneddon , Elements of Partial Differential Equations, McGraw-Hill International Editions (Mathematics Series), 1985
Other Books: R. Dennemeyer , Introduction to Partial Differential Equations and Boundary Value Problems., McGraw-Hill, New York , 1986.
Content : First Order equations, Cauchy-Kovalevsky theorem. Method of characteristics., Lagrange-Charpit method, Pfaff systems. Second order equations, classification, Stokes' and Green's theorems, wave equation, Euler-Darboux equations, Riemann Green function, heat equation, Green's function, Laplace and Poisson's equations, Riesz' method. Please see also the "Course Syllabus" below

Course Schedule

  • Tuesday 08.40-10.30 , SAZ-19
  • Thursday 10.40-12.30, SAZ-19
    Exams
    see previous exams

  • (20%) First Midterm Exam pdf file : October 20 (10.40-12.30 SAZ 19)
  • (20%) Second Midterm Exam pdf file : November 24 (10.40-12.30 SAZ 19)
  • (40%) Final Exam pdf file : December 29, 2005 , 09.00-12.00 (SAZ 04)

  • (20%) Attendance, homeworks .

    Solutions of the first midterm exam first midterm ps file
    Solutions of the second midterm exam second midterm pdf file
    Solutions of the final exam final exam pdf file

    Subjects Covered So far

    (new) 1. September 12
  • Curves and Surfaces in space (R^3)
    Homework Set I
    (new) 2. September 19
  • Methods of solutions of dx/P=dy/Q=dz/R
  • Orthogonal Families Curves
  • Pfaffian Differential forms and Pfaffian Differential Equations
    (new) 3. September 26
  • Solutions of Pfaffian Differential Equations in Three Variables
  • First Order Partial Differential Equations
    (new) 4. Ocober 03
  • First Order Partial Differential Equations
  • Integral Surfaces passing through a given curve
  • Nonlinear Partial Differential Equations of first order
    Homework Set II
    (new) 5. October 10
  • Nonlinear PDEs of First Order
  • Partial Differential Equations of Second Order
    (new) 6. October 17
  • Partial Differential Equations with Constant Coefficients
  • Partial Differential Equations with Variable Coefficients
    Homework Set III
    First Midterm Exam
    (new) 7. October 24
  • Partial Differential Equations with Variable Coefficients
  • Characteristic Curves of Second Order Equations
    (new) 8. November 7
  • The Solution of Linear Hyperbolic Partial Differential Equations (Riemann's Method)
  • Characteristics of Equations in Three or more Variables
  • Separation of Variables
  • The Method of Integral Transform
    Homework Set IV
    (new) 9. November 14
  • The Laplace Equation
  • Laplace equation in spherical, cylindrical and Cartesian coordinates
  • Some simple solutions
  • Equpotential surfaces
    (new) 10. November 21
  • The Laplace Equation
  • Seperation of variables
  • Boundary value problems
  • Kelvin's Theorem
    Second Midterm Exam
    (new) 11. November 29
  • The Laplace Equation
  • The method of Green's function (in two and three dimensions)
  • Some Dirichlet and Neuman problems
  • The wave equation
    Homework Set V
    (new) 12. December 5
  • The wave equation
  • D'Alembert's Solution
  • String with one fixed end
  • String with fixed ends
  • Initial Condtions on a curve (Riemann's Method)
    (new) 13. December 12
  • Membranes
  • Infinite and finite membranes
  • Uniqueness problems
    14. December 19
  • Diffusion Equation
  • The fundemantal solution of the heat equation
  • Solution of the heat equation with several initial and boundary problems,
    15. December 26
  • Maximum and minumum principles of the heat equation
  • Uniqueness of the solutions of the heat equation
  • Similarity solutions of the PDEs F(x,y,u, ux,uy,...)=0
    end of the semester

    Course Syllabus

  • 1. Feb.07 Chapter 1. Section 1.1. Surfaces in three dimensions Solve exercises at the end of the section on page 7.
  • 2. Feb.14 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
    For exercises please see pdeexercises

    For midterms please see pdemidterms

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    Last update January 2002


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