# MTH550 - Introduction to PDEs

IIIT-Delhi

Winter 2020

4 credits

**Instructor**

#### Overview

Partial differential equations are ubiquitous in the sciences and engineering. One encounters them in various contexts : from relatively simple and well studied scenarios such as modeling a vibrating string or membrane using the wave equation to problems as complicated as modeling fluid flows using the Navier Stokes equations : a system that is yet untractable analytically. The present course is an advanced undergraduate / introductory postgraduate course in the subject. It starts with a quick review of first order linear PDEs, classification of second order equations, initial and boundary conditions and the notion of well posed problems. The three most prominent linear PDEs: Wave, Heat and Laplace are then dealt with in detail. Convergence and completeness of Fourier series are studied before taking up a study of harmonic functions. This is followed by a detailed look at Green's identity and Green's functions. Distributions and transforms (Laplace and Fourier) are studied before ending the course with an introduction to nonlinear PDEs wherein we study shock waves and solitons, take a brief look at calculus of variations and bifurcation theory and take up an application to studying water waves.

#### Description

Course Objectives:

CO1: Students will be able to mathematically model physical systems using PDEs (wherever relevant) with an understanding of the physics contained in each term of the PDE.

CO2: Students will be able to determine if a PDE is well posed and solve a range of linear PDEs. They will obtain an in-depth understanding of the conditions required for a solution to exist, the properties it must satisfy, the complexities involved in finding solutions (and hence the need for different methods) and apply the same in solving different PDEs.

CO3: Students will be able to find generalized solutions called distributions as a first approach in some situatons where it is hard to immediately find a solution of a PDE which is at least differentiable.

CO4: Students will be introduced to the role of nonlinearity in PDEs and will see through examples why introduction of nonlinearity may completely alter the behavior of linear solutions.

Lecture Plan:

Week 1: Introduction and examples of PDEs from different areas, Fitst order linear equations, Derivation of some well known PDEs: Transport, Wave, Diffusion, Stationary Wave and Diffusions (Laplace), Initial and Boundary Conditions, Well posed problems, Classification of second order equations, Initial value problem for the wave equation and d'Alembert solution.

Week 2: Wave equation (contd.): Causality and Energy; The diffusion equation, Maximum principle, Uniqueness, Stability, Diffusion on the whole line; Comparison of waves and diffusions, Diffuion on the half line.

Week 3: Reflections of waves, Diffusion with a source, Waves with a source; Separation of variables, The Dirichlet condition.

Week 4: Separation of variables (contd.): The Neumann condition, The Robin condition, Coefficients of Fourier series, Orthogonality and general Fourier series, Convergence of Fourier series.

Week 5:Fourier series (contd.): Completeness, Gibbs phenomenon, Fourier series solutions, Inhomogeneous boundary conditions; Laplace's equation, Maximum principle, Invariance in two and three dimensions.

Week 6: Separation of variables for rectangles and boxes, Poisson's formula, Separation of variables in polar coordinates; Green's first identity, Mean value property, Maximum principle, Uniqueness of Dirichlet and Neumann problems.

Week 7: Green's first identity (contd.): Dirichlet principle; Green's second identity, Representation formula, Green's functions, Symmetry of Green's function, Solution for Green's function for half space and sphere; Waves in space: Energy and Causality, Wave equation in space-time, Rays, singularities & sources.

Week 8: Three dimensional diffusion equation, Schrodiger' equation; Boundaries in the plane and in space: Vibrations of a drumhead, Solid vibrations in a ball, Nodes.

Week 9: Bessel functions, Legendre functions, General eigenvalue problems: Minima of potential energy, Computation of eigenvalues, Commpleteness of eigenfunctions.

Week 10: Symmetric differential operators, Completeness and separation of variables, Asymptotics of eigenvalues.

Week 11: Distributions, Interpreting Green's functions and source functions for Poisson, diffusion and wave equations, Fourier transforms, Finding source functions using Fourier transforms.

Week 12: Laplace transform techniques; Nonlinear PDEs: Shock waves, Solitons.

Week 13: Calculus of variations, Bifurcation theory, Water waves.

#### Evaluation

Two Quizzes: 15% Each

Midsem: 30%

Endsem: 40%

#### Pre-requisites

MTH204 (or equivalent)

#### Class Timings

T, F: 4 - 5:30 PM

#### Office Hours

By Appointment.

#### Textbooks

"Partial Differential Equations: An Introduction" by Walter A. Strauss