A thorough discussion of the class of second-order linear partial differential equations with constant coefficients, in two independent variables. Laplace’s equation, the wave equation and the heat equation in higher dimensions. Theoretical/qualitative aspects: well-posed problems, maximum principles for elliptic and parabolic equations, continuous dependence results, uniqueness results (including consideration of unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations — Green functions, conformal mapping; hyperbolic equations — generalized d’Alembert solution, spherical means, method of descent; transform methods — Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel’s method for inhomogeneous hyperbolic and parabolic equations. [Note: Offered in the Fall of odd years.] Prereq: AMATH 351 and 353; Not open to General Mathematics students
