Credits: 4 (3-1-0)
Description
Linear and semi-linear equations, Cauchy problem, Method of characteristics. Cauchy-Kowalewsky theorem, Holmgren’s Uniqueness Theorem. Classification of second order equations, wave equation in one space dimension, classical and weak solutions, Duhamel’s principle. Laplace equation, fundamental solutions, maximum principles and mean value formulas, Properties of harmonic functions, Green’s function, Energy methods, Perron’s method, Parabolic equations in one space dimension, fundamental solution, maximum principle, existence and uniqueness theorems. Wave equation, Solutions by spherical means, Non-Homogeneous Problems, Duhamel’s principle, Energy Methods. Nonlinear first order PDE’s: Complete integrals, Envelopes and singular solutions. Some special methods for finding solutions: Similarity solutions, Hopf-Cole transformation.