Advanced Partial Differential Equations

The precise idea to study partial differential equations is to interpret physical phenomenon occurring in nature. Most often the systems encountered, fails to admit explicit solutions but fortunately qualitative methods were discovered which does provide ample information about the system without explicitly solving it. In this course we will explore the basic ideas of studying first order equations starting with the inner workings of method of characteristics followed by the three fundamental second order PDE’s namely Laplace equation, Heat equation and Wave equation.
INTENDED AUDIENCE :Graduate students (MSc) and advanced undergraduate.PREREQUISITES : A basic knowledge of several variable calculus is enough.INDUSTRIES SUPPORT :None

COURSE LAYOUT

Week 1-3:Module 1: First order Equations: Method of Characteristics, Transport Equation, Burgers Equation, Riemann problem, Lax-Oleinik Formulae, Entropy Solutions.Week 4-6:Module 2: Laplace Equation: Rotational Invariance and Fundamental Solution, Green’s Function, Mean Value Theorem, Maximum Principle, Liouville theorem, Harnack inequality. Week 7-9:Module 3: Heat Equation: Self-Similarity and Fundamental solution, Maximum Principle, Duhamel’s Principle, Energy Method.Week 10-12:Module 4: Wave equation: D’Alembert’s formulae, Kirchoff Techniques via reflections, Finite speed of Propogation.