Syllabus:
i) Basics of ODE: (local as well as global) existence and uniqueness
results, Picard iteration, Gronwalls inequality, solving some first
order and second order
equations. [7 lectures]
ii) Introduction to PDE: order of a PDE, classification of PDEs into
linear, semilinear, quasi-linear, and fully nonlinear equations,
Examples of equations from
Physics, Geometry, etc. The notion of well-posed PDEs [1 lecture]
iii) First order PDEs: Method of characteristics, existence and
uniqueness results
of the Cauchy problem for quasilinear and fully nonlinear equations. [9
lectures] Second order linear PDEs in two independent variables:
classification
into hyperbolic, parabolic and elliptic equations, canonical forms. [1
lecture]
iv) Laplace equation: Definition of Harmonic functions. Mean-value
property,
Strong Maximum principle for harmonic functions, Liouvilles theorem,
smoothness of harmonic functions, Poissons formula. Harmonic functions
in rectangles, cubes, circles, wedges, annuli. [9 lectures]
v) Heat equation: Fundamental solution of heat equation, Duhamels principle,
weak and strong maximum principles, smoothness of solutions of heat equation,
ill-posedness of backward heat equation. [9 lectures]
vi) Wave equation: well-posedness of initial and boundary value problem
in 1D
and dAlembert formula. method of descent in 2D and 3D. Duhamels
principle, domain of dependence, range of influence, finite speed of
propagation. [8
lectures]
vii) Boundary problems: Separation of variables, Dirichlet, Neumann and Robin
conditions. The method of seperation of variables for Laplace, Heat and Wave
equations. [2 lectures]
Suggested Texts :
(a) Evans, L. C. Partial Differential Equations, AMS, 2010.
(b) Han, Q. A Basic Course in Partial Differential Equations, AMS, 2011.
(c) McOwen, R. Partial Differential Equations: Methods and Applications, Pearson, 2002.
(d) Pinchover, Y. and Rubinstein, J. An Introduction to Partial Differential Equations, Cambridge, 2005.
(e) Fritz John Partial Differential Equations, Springer.
(f) Partial Differential Equations: An introduction by Walter Strauss.
https://www.isibang.ac.in/~adean/infsys/database/MMath/C13PDE.html
- Teacher: Mathew Joseph