Syllabus for Partial Differential Equations Qualifying Exam

The Partial Differential Equations exam covers essentially the material taught in MTH G202 (PDE 1).

Topics

• 1st-order linear, quasi-linear and non-linear PDE’s using the method of characteristics: know how to obtain explicit solutions.
• Classification of 2nd-order linear equations in 2 independent variables: hyperbolic, parabolic and elliptic types.
• Power series solutions and the Cauchy-Kovalevski theorem.
• The wave equation: explicit formulas for the initial-value problem in dimensions 1, 2 and 3; energy and uniqueness; Fourier series solutions; Duhamel’s principle; Huygens’ principle (sharp signals).
• The Laplace equation: Green’s identities, mean value theorem, maximum principle, fundamental solution.
• The heat equation: Fourier series solutions, maximum principle, Gaussian kernel for the pure initial value problem.

References

• Fritz John, Partial Differential Equations, 4th Edition, Applied Mathmatical Sciences, Vol. 1, Springer Verlag, 1995.
• Robert McOwen, Partial Differential Equations: Methods and Applications, 2nd Edition, Prentice Hall, 2002.