# MIT 18 306 - Advanced Partial Differential Equations with Applications (73 pages)

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## Advanced Partial Differential Equations with Applications

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- School:
- Massachusetts Institute of Technology
- Course:
- 18 306 - Advanced Partial Differential Equations with Applications

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MIT OpenCourseWare http 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 02139 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 Coe cients in Linear 1st order pde 01 6 1 9 Statement Discontinuous Coe cients in Linear 1st order pde 02 7 1 10 Statement Discontinuous Coe cients in Linear 1st order pde 03 8 1 11 Statement Discontinuous Coe cients in Linear 1st order pde 04 9 1 12 Statement Discontinuous Coe cients in Linear 1st order pde 05 10 2 Semi Linear First Order PDE 12 2 1 Statement Semi Linear 1st order PDE problem 01 12 2 2 Statement Semi Linear 1st order PDE problem 02 12 1 Rosales 18 306 Problem List 2 3 Kinematic Waves 12 3 1 Statement Conservation Equation in Chromatography 12 3 2 Statement Dispersive Waves and Modulations 13 3 3 Statement Channel Flow 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 In nite conservation laws for kinematic waves 20 3 8 Statement Tra c Flow problem 01 20 3 9 Statement Tra c 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 5 1 27 Statement The importance of being hyperbolic 6 Point Sources and Green functions 27 29 6 1 Statement Green s functions for the wave equation 29 6 2 Statement Cerenkov radiation and Mach cone 35 6 3 Statement Moving point source in 1 D 36 6 4 Statement Nonlinear di usion from a point seed 40 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 Moisture transport in porous media 43 6 4 1 6 4 2 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 48 Rosales 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 56 8 Singularities and characteristics 59 8 1 Statement Singularities in PDE solutions problem 01 59 8 2 Statement Steady State Shallow Water problem 01 59 9 Transformations 60 9 1 Statement Hodograph transformation problem 01 60 9 2 Statement Gas Dynamics Eulerian to Lagrangian formulation 62 10 Wave Equations 64 10 1 Statement Wave equations problem 01 64 10 2 Statement Wave equations problem 02 65 10 3 Statement Wave equations problem 03 66 11 Weak solutions and generalized functions 68 11 1 Statement Weak solutions problem 01 68 11 2 Statement Weak solutions problem 02 69 12 Well and ill posed problems 70 12 1 Statement Ill posed laplacian problem 70 12 2 Statement Laplace equation problem 01 70 12 3 Statement PDE Blow Up 71 List of Figures Rosales 1 1 1 18 306 Problem List 4 Linear First Order PDE Statement Linear 1st order PDE problem 01 Part 1 Find the general solutions to the two 1st order linear scalar PDE x ux y uy 0 and y vx x vy 0 1 1 Hint The general solutions take a particular simple form in polar coordinates Part 2 For u nd the solution such that on the circle x2 y 2 2 it satis es u x Where is this solution determined by the data given Part 3 Is there a solution to the equation for v such that v x 0 x for x Part 4 How does the general solution for u changes if the equation is modi ed to x ux y uy x2 y 2 sin x2 y 2 1 2 1 2 Statement Linear 1st order PDE problem 02 Consider the following problem x u x y uy 1 y 2 with u x 1 1 x for x 1 3 Part 1 Use the method of characteristics to solve this problem Write the solution u u x y explicitly as a function of x and y on y 0 Part 2 Explain why u u x y is not uniquely determined by the problem above for y 0 you may use a diagram 1 3 Statement Linear 1st order PDE problem 03 Consider the following problem ux 2 x uy y with u 0 y f y for y 1 4 where f f y is an arbitrary function Part 1 Use the method of characteristics to solve this problem Write the solution u u x y as a function of x y and f In which part of the x y plane is the solution uniquely determined Part 2 Let f have a continuous derivative Are then the partial derivatives ux and ux continuous Rosales 1 4 18 306 Problem List 5 Statement Linear 1st order PDE problem 04 Discuss the two problems ux 2 x uy y with How many solutions exist in each case a u x x2 1 for 1 x 1 b u x x2 13 x3 for 1 x 1 1 5 Note that the data in these problems is prescribed along a characteristic 1 5 Statement Linear 1st order PDE problem 05 Consider the problem x ux x y uy 1 with u 1 y y for 0 y 1 1 6 Question 1 Write the solution u u x y in the region where it is uniquely determined Question 2 Describe the region in the plane where the solution to 1 6 is uniquely determined Question 3 Write all the functions u u x y that satisfy 1 6 on x 0 and y Question 4 Write all the functions u u x y that satisfy the pde in 1 6 for x 0 Question 5 What happens along x 0 Can you produce solutions to the pde that are continuous in the punctured plane plane minus the origin 1 6 Statement Linear 1st order PDE problem 06 Consider the pde ux 2 …

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