### MA7850 Advanced Differential Equations

#### Course Details

__Ordinary Differential Equations:__

Review of existence and uniqueness of solutions of Initial value problems for system of first order differential equations

Existence and Uniqueness theorem for a linear system; homogeneous and inhomogeneous linear systems; linear equations with constant coefficients; Fundamental matrix

Linear differential equations with periodic coefficients: Floquet theory Stability for linear systems. Principle oflinearised stability. Stability for autonomous systems. l.iapunov functions.

Plane autonomous systems

Periodic solutions of plane autonomous systems.

__Partial Differential Equations:__

Review of method of characteristics for first order partial differential equations and classification of second order partial differential equations Nonlinear first order partial differential equations. Conservation laws. Lax-Oleinik . formula. Riemann's problem.Long time behavior Separation of variables. Similarity solutions. Transform methods. Converting nonlinear partial differential equations into linear partial differential equation. Asymptotics. Maximum principles

#### Course References:

Text Books:

1. R. Grimshaw, Nonlinear ordinary differential equations, Blackwell Scientific publications, 1990.

2. Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, 1991.

Reference:

1. David Betounes, Differential equations: Theory and applications, Springer, 2010.

2. L. Perko, Differential equations and dynamical systems, Springer, 2001.

3. Mathew P. Coleman, An introduction to partial differential equations with Matlab, CRC Press, 2005.

4. Sandro Salsa, Partial differential equations in action: From modelling to theory, Springer, 2008.