Partial Differential Equations for Applications
     110.417MTW  12 - 1 pm,  Krieger Hall,  Room 304
Professor:  Christina Sormani
                          Kriegar 220,  516-6637

Textbook: Elementary Applied Partial Differential Equations,  
               by Richard Haberman, 3rd Edition, Prentice Hall.

Prerequisites:   Multivariable Calculus and Linear Algebra are required.
              Analysis (211.405) is highly recommended especially knowledge of
              convergence of functions and series of functions.    Experience solving
              Ordinary Differential Equations is also highly recommended.

Grading Policy:

       Homework 30%:  All homework which is assigned is due the following Monday .
              It is expected that some reading and problems will be done Mondays and Tuesdays.
              More homework will be assigned during the first half of the semester than the later
              half and will be weighted accordingly.

       Midterm 20%  This exam will be given in the fifth week of class.  It will consist
              of problems similar to homework problems and some multiple choice.

       Project 20%:  This group project can be worked on with up to three students,
             with the quantity of work reflecting the number of students involved.
             It should consist of a numerical solution of a PDE using any computer language
             the students choose.   It should also include an analysis of the convergence
             of the solution.  Students who haven't used computers in the past should work
             with those who have.
       Final 30%:    This exam will be given during the finals week.  It will consist of
              problems similar to homework problems, some multiple choice on general
              knowledge of PDEs, and a question or two regarding the project.

Syllabus:   (This syllabus will be updated as the course progresses)
        Week 1:  Introduction to PDEs
                           Derivation of the Heat Equation  (1.2 ,1.5)
                           Boundary Conditions and Steady State Solutions  (1.3,1.4)
                 HW:   Steady State Solutions, Review Gradients
                            Handout #1 1-7,  1.2.3, 1.5.8, 1.4.1, 1.4.4, 1.4.6, 1.4.7 (a),
                           1.5.3,  1.5.8, 1.5.14, 1.5.19, 1.5.20

        Week 2:   Real Vector Spaces (Defn, Spanning, Linear Indep, Dimension)
                            Linear Operators  (2.2)
                            Separation of Variables (2.3.2-2.3.3)
                  HW:   Handout #2, 2.2.1, 2.2.2, 2.2.4, 2.2.5, 2.3.1
        Week 3:   Pointwise Convergence of Fourier Series (Handout #3, 2.3.4, 2.3.5-6)
                            Integrating Series, L2   Inner Product  (2.3.6, appendix to 2.3, Handout #4),
                            Self Adjoint Operators and Eigenfunctions (Handout #3, 2.4)
                 HW:    Read all of 2.1-2.4 (covers explicit techniques used to solve problems)
                            Do: Handout #3, Handout #4, 2.3.3 ab, 2.3.5, 2.3.7, 2.3.10abc,
                            2.4.1 abcd (verify convergence of these Fourier Series), 2.4.3
        Week 4:   President's Day
                            The Convergence Theorem of Fourier Series (3.2)
                            The Gibbs Phenomenon  (3.3.1) and Uniform Conv  (Handout #5)
                  HW:   Handout #5 1.2/ 2,5,8, 9ab,10   Read through 1.3.4
                            Text read 3.1-3.3, do 3.2.1, 3.2.2, 3.2.4, 3.3.1, 3.3.7
                  Extra Credit :  Graph the Fourier Series up to 10 terms for various functions
                            using Mathematica or some other language to illustrate uniform

       Week 5:   Uniform Conv, (3.3-3.5, Handout #5)
                           Estimating and Maximizing Solns of the Heat Eqn (Handout #6)
                           Differentiation and Integration of  Series (3.3.2-3.3.4)
                  HW:   Read 3.3.2-3.3.4 ,  Examples
        Week 6:    Midterm Exam on 1.1-1.5, 2.1-2.4, 3.1-3.5, Handouts #1-#6
                              Laplace's Equation and the Maximum Principle (2.5.1-2)
                              Laplace's Eqn on the disk and the Mean Value Prop (2.5.3-2.5.4)
                  HW:     Handout #7, 1-3, 2.5.2, 2.5.10, 2.5.11, 2.5.12, 2.5.13, 2.5.14
        Week 7:    Vibrating Strings and Membranes (4.2, 4.5)
                              Travelling Waves, Dependence and D'Alembert Solution (12.3)
                              Travelling Waves and Boundary Data  (12.4-5)
                 HW :  12.3.1, 12.3.4, 12.3.5, 12.4.1, 12.4.2,12.4.4 , 4.4.9, 4.4.11,
                            4.4.12, Handout #8 on uniqueness of solutions of the wave eqn.
                            Read Chapter 4 on solving the wave eqn with Fourier Series.
      Spring Break

        Week 8:   Fourier Series and Waves (4.4)
                            Fourier Series and Mathematica   (Solns for Handout #7)
                            Finite Difference Equations  (6.1-6.2)
                HW :    Handout #9 using mathematica (may take a lot of time),
                            12.3.3, 12.3.6 , 6.2.1,

         Week 9:   Numerically Solving the Heat Eqn, (6.3.2)
                             Propogation of Disturbances  (6.3.3)
                             Difference Equations (Handout #10, 6.3.5, 6.3.6)
                  HW:   6.2.3, 6.2.4, Read Handout #10 and consult
                            Sections 6.3.4-6.3.6 (which are a bit confusing)
                            Do problems 1-10 of the handout.
                            Choose project partners.

         Week 10: Stability Analysis  and Lax Equivalency Theorem (Handout #10, 6.3.4)
                              Other Numerical Schemes (6.3.7-6.3.9) (Skip Random Walks)
                              Two Dimensional Heat Equation and Stability (6.4)
                   HW:    Problems 11-13 of Handout #10 , 6.2.6, 6.2.7, 6.3.1, 6.3.3, 6.3.6
                               6.3.12, 6.3.13  Write up the Project's Aims
         Week 11: Neumann Boundary Conditions (6.3.9)
                              Wave Equation solved Numerically (6.5)
                              Laplace's Equation with Jacobi and Gauss-Seidel (6.6)
                   Project HW:    Write out the exact PDE including boundary conditions
                               for the project.  Be specific about all known functions
                               and unknown functions.  Write up a consistent difference
                               equation (for PDE and boundary data and initial data)
                                and find the truncation errors.
                               Verify stability of this equation and discuss appropriate
                               grid ratios.  Check that the truncation errors are the
                               same size given these grid ratios.  If not, make better
                               approximations where appropriate and repeat the process.
                               Hand in all work.   If your partners are not available, do this
                               yourself and hand it in with your own name on it.
                    HW:   6.5.3, 6.5.4, 6.5.5, 6.5.6, 6.6.3 a,b
                    Extra Credit: 6.5.2, use any programming language you wish.

       Week 12:  Laplace's Equation with S-O-R iteration(6.6)
                             Von Neumann Stability with Complex eigenvalues
                             Hints for various Projects
                     HW:   6.6.1, 6.6.3c, 6.6.4, and 6.6.5
                     Project HW:  Write the computer program for your project and debug it!
      Week 13:  Sturm-Liouville and Helmholtz Eigenvalue Problems:
                                 Part I: Green's Formula and Orthogonality (5.2-5.3.3, 7.4)
                                 Part II :  Uniqueness and Eigenfunctions (5.5, 7.5)
                                 Part III: Rayleigh Quotient  (5.6, 7.6)
                    HW:   Fill out the Teaching Evaluations at
                                 Read 7.3 for the Helmholtz Equation on a rectangle
                                 5.3.1, 5.4.2, 5.5.11 (reviews self adjointness Handout #3)
                                 7.4.1, 7.5.1a, 7.5.2a, 7.6.1, 7.6.2a, 7.5.4
                    Please show me some code for your project during office hours Friday
                                 if not before.  I can help catch bugs.

    Week 14:  Heat Flow in a Nonuniform Rod and a Survey of PDE  (5.4)
                               Review  Session May 7,  6-7pm  Room 211 Krieger Hall
                               Projects are Due May 7, 6pm (Each student should keep a
                               copy  of the project to study for the final)
         Final Exam:  Tuesday May 12, 2-5 pm, Room 304 Krieger Hall
                         Covers all sections and Handouts listed above.
                         Types of questions that might be asked:
                                  Linear Differential Equations, Homogeneous Equations
                                              Identify, verify
                                  Seperation of Variables
                                              find product solutions,
                                              maybe seperate 3 variables)
                                  Fourier Series, Gibbs, (as in exam I)
                                  Uniform Convergence
                                              verify that something converges uniformly 
                                  Max principle, Mean Value Property
                                               prove uniqueness,
                                               identify solutions of Laplace's eqn given their graphs
                                  Charateristics, reflection...
                                               solve a given wave eqn using this method
                                               verify that a soln of a wave equation can be found this way
                                  Find a consistant difference equation
                                                verify using Taylor series
                                  Check stability for time dependant difference equation
                                  Numerics for Laplace's equation
                                                discuss relation to mean value prop
                                  Eigenfunctions and solving the heat equation
                                                 given eigenfunctions find solution using series,
                                                 verify orthogonality
                                  Eigenvalues, Rayleigh quotient
                                                 check if positive, give example where evalue isn't positive
                                                 find eigenvalue for a given eigenfunction
                                  Self Adjoint Operators
                                                 verify a given linear operator is self adjoint
                                                     (by integrating by parts or using Green's formula)