MIT OpenCourseWarehttp://ocw.mit.edu 18.306 Advanced Partial Differential Equations with Applications Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.306 Problem List. Rodolfo R. Rosales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02 139 March 19, 2008 Abstract Problem list for 18.306. These problems may be assigned in problem sets and/or exams. Contents 1 Linear First Order PDE. 4 1.1 Statement: Linear 1st order PDE (problem 01) . . . . . . . . . . . . . . . . . . . . 4 1.2 Statement: Linear 1st order PDE (problem 02) . . . . . . . . . . . . . . . . . . . . 4 1.3 Statement: Linear 1st order PDE (problem 03) . . . . . . . . . . . . . . . . . . . . 4 1.4 Statement: Linear 1st order PDE (problem 04) . . . . . . . . . . . . . . . . . . . . 5 1.5 Statement: Linear 1st order PDE (problem 05) . . . . . . . . . . . . . . . . . . . . 5 1.6 Statement: Linear 1st order PDE (problem 06) . . . . . . . . . . . . . . . . . . . . 5 1.7 Statement: Linear 1st order PDE (problem 07) . . . . . . . . . . . . . . . . . . . . 5 1.8 Statement: Discontinuous Coefficients in Linear 1st order pde #01 . . . . . . . . . . 6 1.9 Statement: Discontinuous Coefficients in Linear 1st order pde #02 . . . . . . . . . . 7 1.10 Statement: Discontinuous Coefficients in Linear 1st order pde #03 . . . . . . . . . . 8 1.11 Statement: Discontinuous Coefficients in Linear 1st order pde #04 . . . . . . . . . . 9 1.12 Statement: Discontinuous Coefficients in Linear 1st order pde #05 . . . . . . . . . . 10 2 Semi-Linear First Order PD E. 12 2.1 Statement: Semi-Linear 1st order PDE (pro blem 01) . . . . . . . . . . . . . . . . . 12 2.2 Statement: Semi-Linear 1st order PDE (pro blem 02) . . . . . . . . . . . . . . . . . 12 1Rosales 18.306 Problem List. 2 3 Kinematic Waves. 12 3.1 Statement: Conservation Equation in Chromatography . . . . . . . . . . . . . . . . 1 2 3.2 Statement: Dispersive Waves and Modulations . . . . . . . . . . . . . . . . . . . . 13 3.3 Statement: Channel F low Rate Function . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Statement: Road capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5 Statement: Initial Values for a Kinematic Wave (problem 01) . . . . . . . . . . . . . 16 3.6 Statement: Initial Values for a Kinematic Wave (problem 02) . . . . . . . . . . . . . 17 3.7 Statement: Infinite conservation laws for kinematic waves . . . . . . . . . . . . . . . 20 3.8 Statement: Traffic Flow problem 01. . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.9 Statement: Traffic Flow problem 02. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.10 Statement: Envelopes and cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Hamilton Jacobi and Eikonal Problems. 25 4.1 Statement: Eikonal equation (problem 01) . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Statement: Eikonal equation (problem 02) . . . . . . . . . . . . . . . . . . . . . . . 27 5 HyperbolicEquations. 27 5.1 Statement: The importance of being hyperbolic . . . . . . . . . . . . . . . . . . . . 27 6 Point Sources and Green functions. 29 6.1 Statement: Green’s functions for the wave equation . . . . . . . . . . . . . . . . . . 29 6.2 Statement: Cerenkov r adiation and Mach cone . . . . . . . . . . . . . . . . . . . . . 35 6.3 Statement: Moving point source in 1-D . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.4 Statement: Nonlinear diffusion from a point seed . . . . . . . . . . . . . . . . . . . 40 6.4.1 Example: Green function for the heat equation in Rd . . . . . . . . . . . . . 42 The area of a sphere in d-dimensions . . . . . . . . . . . . . . . . . . . . . . 43 Initial value for the linear heat equation: existence and uniqueness . . . . . 43 6.4.2 Moisture transport in porous media . . . . . . . . . . . . . . . . . . . . . . . 43 7 Shock Jump and Entropy Conditions. 44 7.1 Statement: Lax entropy cond. for scalar convex cons. laws, and inform. loss . . . . 44 7.2 Statement: Zero viscosity limit in scalar convex cons. laws and dissipation . . . . . 48Rosales 18.306 Problem List. 3 7.3 Statement: Entropy conditions for scalar (non convex) problems . . . . . . . . . . . 50 7.4 Statement: Gas Dynamics strong shock conditions . . . . . . . . . . . . . . . . . . . 53 7.5 Statement: Entropy conditions for the p-system . . . . . . . . . . . . . . . . . . . . 54 7.6 Statement: Shallow water - Energy dissipation at shocks . . . . . . . . …