Integrated MScPhD Program SYLLABUS 
KSM4E03: Partial Differential Equations 
Topics covered: Order of a PDE, classification of PDEs into linear, semilinear, quasilinear, and fully nonlinear equations, examples of equations from physics, geometry, etc., the notion of wellposed 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, meanvalue 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 . Weak maximum principle, heat mean value formula, strong maximum principle, smoothness of solutions of heat equation, illposedness of backward heat equation, energy methods. Wave Equation: Wellposedness 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 multidimension, notion of real analytic function and its properties. CauchyKowalevski theorem, Holmgren uniqueness theorem. Limitations of Classical Solutions and Weak Solutions: Lewy’s example of PDE for which no solution exists, HamiltonJacobi equations, conservation laws. Suggested texts:
