Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Exam Number 02
18.306 MIT (Fall 2009)
Rodolfo R. Rosales
December 2, 2009
Due: Last day of lectures.
Contents
1.1 Statement: Greens functions (problem 01).
.
1
Greens function for the heat equation in the innite line. . . . . . . . . . . . . .
1
1.2 Sta
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
MIT 18.306 Fall 2009
Problem set #1
Due Friday October 2
Problem #1
Sove the following problem, using the method of characteristics (compute
the characteristics as done in the lectures, then solve for the solution
along the characteristics, and then elimi
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 07 2009 09 30 WED
TOPICS: Region of multiple values. Envelope of characteristics.
Continue with u_t + c(u)*u_x = 0 and u(x, 0) = F(x).
Study boundary of the region of multiple values. Show that this
is equivalent (as long as dc/du never vanishes)
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 02 2009 09 14 MON
TOPICS: Conservation laws and pde.
Integral and differential forms.
Closure strategies. Quasiequillibrium.
Derivation of pde by conservation laws. Integral and differential
forms.
 The pde given by a conserved density and the c
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 08 2009 10 05 MON
TOPICS: More on envelopes. Infinite slopes at envelope.
Shocks. Conservation and entropy. Irreversibility.
Examples from traffic flow.
Continue with c_t + c*c_x = 0 and c(x, 0) = C(x).
Show alternative definition of envelope of a
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 11 2009 10 14 WED
TOPICS: The Riemann problem for the kinematic wave equation
with convex/concave flux.
Example of a conservation law with a point source term.
Riemann problem for:
Case Traffic Flow
Case River
Flows
u_t + Q(rho)_x = 0
Q concave
Q
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 03 2009 09 16 WED
TOPICS: Classification of pde.
Examples.
Kinematic waves and characteristics.
Definition of PDE. Rank PDE from general to simplest.
Quasilinear, semilinear, linear, high order, first order, systems,
scalar .
Simplest pde: scala
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 10 2009 10 13 TUE
TOPICS: Shocks in the presence of source terms. Example.
Riemann problems and Godunov's type methods.
Shocks for equations with source terms.
Example:
u_t + (0,5*u^2)_x = 1.
Study characteristics, crossings and shock formation.
D
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 04 2009 09 21 MON
TOPICS: First order scalar pde.
Examples of solutions by characteristics.
Domain of influence.
Review characteristics.
Examples in detail:
1) x*u_x + y*u_y = 0,
for y >= 1, with u(x, 1) = F(x)
2) x*u_x + y*u_y = 1+y^2,
for y >= 1
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 05 2009 09 23 WED
TOPICS: Domains of influence and dependence.
Causality and uniqueness. Allowed boundary conditions.
Examples.
Domain of definition and domain of dependence: where is the solution
defined.
Implications for where conditions must be
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 01 2009 09 09 WED
TOPICS: Mechanics of the course.
Example pde. Initial and boundary value problems.
Well and illposed problems.
Introduction: Syllabus issues; exams; lecturer; etc.
ODE's and PDE's
ODE solution: determined by a set of constants.
Advanced Partial Differential Equations with Applications
MATH 18.306

Fall 2009
Lecture 06 2009 09 28 MON
TOPICS: Graphical interpretation of solution by characteristics.
Conservation. Wave steepening and breaking.
Back to the physics.
Continue with u_t + c(u)*u_x = 0 and u(x, 0) = F(x).
Graphical interpretation of the solution by ch