This topic includes: Linearity and superposition principle, classification of linear partial differential equations of second order; one dimensional wave equation, initial value problem, clamped and free boundary conditions, non-homogeneous wave equation, uniqueness and stability of solution; three-dimensional wave equation, Kirchhoff's formula; heat equation, initial value problem, clamped and free boundary conditions, the maximum principle, uniqueness and stability of solution; three-dimensional heat equation. Laplace equation, Green's formulas, Dirichlet and Neumann problems, the maximum principle and uniqueness of solution; Green's function. Fourier transform in n dimensions, generalized functions and generalized solutions of equations. Fredholm theory of integral equations.
This topic aims to equip students with the skills needed to solve mathematical problems in the theory of partial differential equations. The focus is on both the application of mathematical ideas to practical problems and on rigorous understanding of mathematical principles.