The course covers
the following topics: single First-Order
equation, the Cauchy problem, systems
of first-order equations, characteristics,
the Cauchy-Kowalevski Existence and
Uniqueness Theorem, elliptic equations,
the Laplace equation, the Lagrange-Green
identity, the maximum principle, harmonic
and subharmonic functions, Green’s
function and the Poisson formula, hyperbolic
equations in higher dimensions, the wave
equation, the method of Spherical means,
Hadamard’s method of descent, hyperbolic
equations with constant coefficients,
solution by the n-dimensional Fourier
Transform, Parabolic equations, and the Heat
equation. *Prerequisite:* Graduate Standing.