Partial Differential Equations (PDEs) appear as mathematical models for many a physical phenomena. Closed-form solutions to most of these PDEs cannot be found. One of the possible ways to understand the models is by studying the qualitative properties exhibited by their solutions.In this course, we study first order nonlinear PDEs, and the properties of the three important types of second order linear PDEs (Wave, Laplace, Heat) would be studied and compared.INTENDED AUDIENCE :Mathematics, Physics, Mechanical Engineering, Chemical EngineeringPRE-REQUISITES : Exposure to Multivariable calculus is needed. In addition, exposure to Linear algebra would be ideal.INDUSTRY SUPPORT : NILL
Week 1:Introduction, First order partial differential equations, Method of characteristicsWeek 2:Cauchy problem for Quasilinear first order partial differential equationsWeek 3:Cauchy problem for fully nonlinear first order partial differential equationsWeek 4:Classification of Second order partial differential equations and Canonical formsWeek 5:Wave equation: d’Alembert’s formula, Solution of wave equation on bounded domainsWeek 6:Wave equation: Solution by method of separation of variables, Wave equation in two and three space dimensionsWeek 7:Wave equation: Parallelogram identity, Domain of dependence, Domain of influence, Causality principleWeek 8:Wave equation: Finite speed of propagation, Conservation of energy, Huygens principle, Propagation of confined disturbancesWeek 9:Laplace equation: Boundary value problems, Fundamental solution, Construction of Greens function for Dirichlet problem posed on special domains.Week 10:Laplace equation: Poisson’s formula, Solution of Dirichlet problem on a rectangle by method of separation of variables Week 11:Laplace equation: Mean value property, Maximum principles, Dirichlet principleWeek 12:Heat equation: Fundamental solution, Solution of initial-boundary value problem by separation of variables method, Maximum principle.