Math 294:      Partial Differential Equations 
Lectures:      Scheduled:  MWF 11:10am-12:00noon
Professor:     Dr. Mary Ann Horn 
Email:         horn@math.vanderbilt.edu
Office:        1414 Stevenson Center, 343-5905 
Text:          A First Course in Partial Differential Equations, H. Weinberger 


     An introduction to methods of partial differential equations, this
course will include such topics as orthogonal systems and Fourier series;
derivation and classification of partial differential equations; maximum 
principles; eigenvalue function method and its applications; Green's 
functions; as well as additional topics.  Prerequisites include 
a course in ordinary differential equations (e.g. Math 198 or Math 208).
While the focus will be on techniques and theory for one, two, and three-
dimensional problems on ``nice'' domains, I hope to motivate some of the 
material through discussion of modern applications and research topics.

Professor Weinberger sums up his motivation as to why partial differential
equations should be studied in his introduction to his book, stating that,
``It has become customary at many colleges and universities to teach
undergraduate courses in boundary value problems, Fourier series, and
integral transforms.  These courses usually emphasize the Fourier
series or Laplace transforms, and then treat some problems in partial
differential equations as applications.

Those who are mathematically inclined are left with the impression that
the solution of partial differential equations consists of some rather dull
manipulations with infinite series or integrals, and is not worthy of 
further study.  Those students who are primarily interested in technical 
applications also get the feeling that all partial differential equations 
can be treated by separation of variables or integral transforms.  When a 
problem arises to which such methods do not apply (and this often happens 
quite soon), they either use the only methods they know, arriving at wrong 
results, or they simply give up in exasperation.  After this, they have a 
strong feeling that the mathematicians have cheated them, and tend to 
distrust all mathematical techniques.
 
     This ... is an attempt to present materials usually covered in such
courses in a framework where the general properties of partial differential 
equations such as characteristics, domains of dependence, and maximum 
principles can be clearly seen."

     I hope to leave you with both an intuitive feel for how the solutions
are behaving and some knowledge of how to apply various techniques to 
find explicit solutions and when not to even try.  Following is a section 
of the contents of the text which may give you an idea of the topics that 
will be discussed during the class.  Chapters I through V will definitely 
be covered.  Chapter VII will depend on the amount of time remaining.



                           Contents

I.  The one-dimensional wave equation 

    1. A physical problem and its mathematical models; the vibrating string 
    2. The one-dimensional wave equation 
    3. Discussion of the solution:  Characteristics 
    4. Reflection and the free boundary problem 
    5. The nonhomogeneous wave equation 

II.  Linear second-order partial differential equations in two variables 

    6. Linearity and superposition 
    7. Uniqueness for the vibrating string problem 
    8. Classification of second-order equations with constant coefficients 
    9. Classificiation of general second-order operators 

III.  Some properties of elliptic and parabolic equations 

    10. Laplace's equation 
    11. Green's theorem and uniqueness for the Laplace's equation 
    12. The maximum principle 
    13. The heat equation 

IV.  Separation of variables and Fourier series 

    14. The method of separation of variables 
    15. Orthogonality and least squares approximation 
    16. Completeness and the Parseval equation 
    17. The Riemann-Lebesgue lemma 
    18. Convergence of the trigonometric Fourier series 
    19. Uniform convergence, Schwarz's inequality, and completeness 
    20. Sine and cosine series 
    21. Change of scale 
    22. The heat equation 
    23. Laplace's equation in a rectangle 
    24. Laplace's equation in a circle 
    25. An extension of the validity of these solutions 
    26. The damped wave equation 

V.  Nonhomogeneous problems 

    27. Initial value problems for ordinary differential equations 
    28. Boundary value problems and Green's function for ordinary 
        differential equations 
    29. Nonhomogeneous problems and the finite Fourier transform 
    30. Green's function 

VII.  Sturm-Liouville theory and general Fourier expansions 

    36. Eigenfunction expansions for regular second-order ordinary 
        differential equations 
    37. Vibration of a variable string 
    38. Some properties of eigenvalues and eigenfunctions 
    39. Equations with singular endpoints 
    40. Some properties of Bessel functions 
    41. Vibration of a circular membrane 
    42. Forced vibration of a circular membrane:  Natural frequencies 
        and resonance 
    43. The Legendre polynomials and associated Legendre functions 
    44. Laplace's equation in the sphere 
    45. Poisson's equation and Green's functions for the sphere