Unit-I

Review of first order equations and characteristics.
Weak solutions to hyperbolic equations- discontinuous solutions, shock formation, a formal approach to weak solutions, asymptotic behaviour of shocks.

Unit-II

Diffusion Processes-Similarity methods, Fisher’s equation, Burgers’ equation, asymptotic solutions to Burgers’ equations.

Unit-III

Reaction diffusion equations-traveling wave solutions, existence of solutions, maximum principles and comparison theorem, asymptotic behaviour.

Unit-IV

Elliptic equations-Basic results for elliptic operators, eigenvalue problems, stability and bifurcation.

Unit-V

Hyperbolic system.

CO1- Understand the general concept of weak solution and the criterion of having weak solution for hyperbolic equation.
CO2- Able to model the basic diffusion processes and understand the mathematical methods that are useful in studying the structure of their solutions.
CO3- Understand the existence and uniqueness of traveling wave solutions solutions.
CO4- Understand the concept of nonlinear eigenvalue problem the stability of equilibrium solutions for reaction-diffusion equation.
CO5- Understand the formulation of system of PDEs and their applications.

• J David Logan, An Introduction to Nonlinear Partial Differential Equations, John Wiley and Sons, Inc., 1994