Course information

First-order Differential Equations: The quasi-linear equation, Cauchy problem, existence, uniqueness, initial value problems. Laplace Equation: Harmonic functions, maximum principle, uniqueness of solutions for Dirichlet – Neumann boundary value problems. Introduction to the theory of generalized functions: Definition, distributions derived from integrable functions, Dirac function, Heaviside function, generalized derivatives. Integral representations of solutions for the Laplace equation: Fundamental solution for the Laplace equation, Green s function, method of images for half-planes and half-spaces, disks and spheres. Wave Equation: Introduction to wave propagation, plane and spherical waves, initial and boundary value problems, characteristic cone and energy theorems, the method of spherical means and Huygens principle. Heat Equation: The initial value problem, the maximum principle, uniqueness and regularization of solution.