Most of the topics of this exam are covered in the following courses: Mathematics in Modern Science I (Mate 6675) and Elementary Partial Differential Equations (Mate 6677). Topics:
- Theory of Fourier series and mathematical validation of the method of separation of variables: Convergence, term-by-term differentiation and integration of Fourier series, applications of the latter to boundary value problems for heat, wave and Laplace equations.
- Fist order quasi linear equations: Linear transport equation, Burgers equation and method of characteristics.
- Elementary distribution theory: Test functions and convergence, distributions represented by locally integrable functions and other important distributions, operations on distributions, fundamental solutions of differential operators.
- Heat (Diffusion) equation: Fundamental solution, initial value problem and uniqueness, Duhamel’s principle, maximum principle.
- Laplace equation: Harmonic functions, Green’s formula, mean value theorem, maximum principle, Dirichlet and Neumann boundary value problems and uniqueness property, Green’s functions.
- Wave equation: Initial value problem and uniqueness, Duhamel’s principle, d’Alembert formula: domain of dependence, region of influence.
Bibliography:
- Introduction to Partial Differential Equations and Hilbert Space Methods, Gustafson, John Wiley and Sons, 1997.
- Applied Partial Differential Equations with Fourier series and Boundary Value Problems, R. Haberman, Pearson, 2013.
- Partial Differential Equations in Action from Modeling to Theory, S. Salsa, Springer, 2008.
- Introduction to Partial Differential equations, G. Folland, Princeton University Press, 1995.
- Partial Differential Equations, Fritz John, Springer Verlag, 1982.
- Partial Differential Equations, J. Rauch, Springer Verlag, 1991.
- Partial Differential Equations, Lawrence C. Evans, AMS, 2000.
- Partial Differential Equations: An Introduction to Partial Differential Equations, W. A. Strauss, 2007