Integrated MSc-PhD Program


KSM4E03: Partial Differential Equations

Topics covered: Order of a PDE, classification of PDEs into linear, semi-linear, quasilinear, and fully nonlinear equations, examples of equations from physics, geometry, etc., the notion of well-posed PDEs.

First Order PDEs: Method of characteristics, existence and uniqueness results of the Cauchy problem for quasilinear and fully nonlinear equations.

Second Order Linear PDEs in Two Independent Variables: Classification into hyperbolic, parabolic, and elliptic equations, canonical forms, the method of separation of variables for Laplace, heat, and wave equations.

Laplace Equation: Definition of harmonic functions, mean-value property, strong maximum principle and Harnack inequality for harmonic functions, Liouville theorem, smoothness of harmonic functions, fundamental solution, Green’s function, examples for Green’s functions like upper half space and ball, Poisson’s formula, energy method, Dirichlet principle, uniqueness using energy method.

Heat Equation: Fundamental solution of heat equation, Duhamel’s principle and formula for solution of u_t - \Delta u = f(x,t), u(x,0)= g(x). Weak maximum principle, heat mean value formula, strong maximum principle, smoothness of solutions of heat equation, ill-posedness of backward heat equation, energy methods.

Wave Equation: Well-posedness of initial and boundary value problem in 1D and d’Alembert’s formula, method of descent in 2D and 3D, Duhamel’s principle, domain of dependence, range of influence, and finite speed of propagation, energy method.

Real Analytic Theory: Definition of power series in multi-dimension, notion of real analytic function and its properties. Cauchy-Kowalevski theorem, Holmgren uniqueness theorem.

Limitations of Classical Solutions and Weak Solutions: Lewy’s example of PDE for which no solution exists, Hamilton-Jacobi equations, conservation laws.

Suggested texts:

            1. Partial Differential Equations by L. C. Evans
            2. A Basic Course in Partial Differential Equations by Q. Han
            3. Partial Differential Equations: Methods and Applications by R. McOwen
            4. An Introduction to Partial Differential Equations by Y. Pinchover and J. Rubinstein
            5. Partial Differential Equations by Fritz John