First part of a sequence of courses on partial differential equations (PDE) emphasizing theory and solution techniques for linear equations. Origin of PDE in science and engineering. Equations of diffusion, heat conduction, and wave propagation. The method of characteristics. Classification of PDE. Separation of variables, theory of the Fourier series and Fourier transform. The method of Green’s functions. Sturm- Liouville problem, special functions, eigenfunction expansions. Higher dimensional PDE and their solution by separation of variables, transform methods, and Green’s functions. Introduction to quasi-linear PDE and shock waves. AMCS 231 was previously entitled "Applied Partial Differential Equations" during Falls of 2009, 2010, 2011, 2012, 2013 and 2014.