Geometrical interpretation of a first-order pde, method of characteristics and general solutions, Monge cone, Lagrange’s equations, canonical forms of first-order linear equations, method of separation of variables.

Second-order equations in two independent variables, canonical forms, equations with constant coefficients, general solutions. 

The Cauchy problem, the Cauchy-Kowalewskaya theorem, homogeneous wave equations, the D’Alembert solution of wave equation, initial boundary-value problems, equations with nonhomogeneous boundary conditions, vibration of finite string with fixed ends,.(review) nonhomogeneous wave equations. 

Basic concepts, types of boundary-value problems, maximum and minimum principles, uniqueness and continuity theorems. Dirichlet problem for a circle, Dirichlet problem for a circular annulus, Neumann problem for a circle, Dirichlet problem for a rectangle, Dirichlet problem involving the Poisson equation, the Neumann problem for a rectangle. 

Derivation of the heat equation and solutions of the standard initial and boundary value problems, uniqueness and the maximum principle, time-independent boundary conditions, time-dependent boundary conditions.

Text Books

1. Tyn Myint-U, Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, Birkhauser,Boston, Fourth Edition, 2007. 
2. D. Bleecker, G. Csordas, Basic Partial Differential Equations, Van Nostrand Reinhold, New York, 1992. 

References:

1. L.C. Evans, Partial Differrential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, Providence, 1998. 
2. I.N. Sneddon, Elements of part