Overview of PDEs

Second-order linear PDEs in two variables with real coefficients can be classified as parabolic, elliptic, or hyperbolic. It is possible to extend this classification to the modes of systems of linear and nonlinear PDEs in two or more variables. Dirichlet, Neumann, and Robin boundary conditions are prescribed for parabolic and elliptic PDEs, while Cauchy boundary conditions are prescribed for hyperbolic PDEs. A method for classifying general nonlinear systems of PDEs is given, using the symbol of the linearized PDE system. A proof of the Equivalence Theorem for PDE initial value problems is given at the end of the chapter.