Math 475B - 31 March 11
Page 1 of 5 - Mar 31, 2011
More on homework 4
Problem 3- how to include parameters
Partial differential equations
The model problems
Discretizing using nite differences
HW 4- problem 3
How to change B0 in command line, passing in
Math 475B - 3 March 11
Page 1 of 6 - Mar 3, 2011
Go over exam
Finish BVP shooting set up
HW3
Exam:
1) recognize that the problem is stiff, notice that there were two
different time scales, -1e6 and (-cos(t) - sin(2t).
-Could also compare a non stiff sol
Math 475B - 29 March 11
Page 1 of 5 - Mar 29, 2011
BVPs in Matlab
How to give a constant guess
Homework 4
Solving BVPs in Matlab
Bvp5c (different error control)
BVP4c (rst choice) - adaptive, takes your guess and nds a solution
as well as a mesh a<x1<x2
Math 475B - 27 Jan 11
HW: 7.1!
!
! express y' in terms of y and t!
! create a new dependent
7.3!
! error estimator, 7.5, e = error estimator!
!
! Show that BS23 is exact for 1,t,t^2 but not t^3!
! ! Experimentally: use matlab, call ode23tx!
! ! How to sho
Math 475B - 13 Jan 11
Page 1 of 5 - Jan 13, 2011
Solving (integrating) differential equations numerically
Mostly numerical methods
Grade: 25% each for two exams, 25% homework, 25% nal project (no
nal exam)
Open laptop exams
Final Project
-Project that do
Math 475B - 18 Jan 11
Page 1 of 5 - Jan 18, 2011
Euler contd
that gives the Forward Euler
But if we use right hand sums
This gives us a way to solve for the y(t+h) in terms of y(t) via an
implicit equation
Newton's method: linearize equation, solve linear
Math 475B - 20 Jan 11
Page 1 of 5 - Jan 20, 2011
more about order (in practice)
list of more ODE solving methods
Properties of solvers:
xed/adaptive step size
Order
explicit/implicit
single/multi step (to be dened)
linear/nonlinear
Using Matlab's bui
Math 475B - 25 Jan 11
Newton's Method: used to nd the roots of some function
ex:
need:
1) an initial guess
2) function
3) derivative of the function
new guess
in general
Page 1 of 5 - Jan 25, 2011
Math 475B - 25 Jan 11
Page 2 of 5 - Jan 25, 2011
newton
Math 475B - 3 Feb 11
Agenda:
1) (see ch odes, annotated)
2) discuss 7.6 hw
3) iftime
-linearization, diagonalization, test
Note for varargin (see ode23tx annotated notes)
function ydot = stiff ( t, y , lambda, A)
ydot = - lambda(y-sin(t) + Acos(t);
e
Math 475B - 17 Feb 11
Page 1 of 2 - Feb 17, 2011
Exam
Open book, open laptop, open notes, web ok
What is an IVP?
What does the order of a method mean?
Matlab problems will be similar to homework 1 and 2
For sure: sensitivity analysis on a simpler sys
Math 475B - 22 March 11
Page 1 of 7 - Mar 22, 2011
Outline due Tuesday 3/29
Review shooting
Multiple shooting
Finite differences
Using Matlab's built in BVP4C
In the shooting method, we use an IVP solver together with Newton's
Method and a good guess to n
Math 475B - 1 March 11
Page 1 of 6 - Mar 1, 2011
Boundary Value Problems:
examples
Standard form
shooting method
HW3
BVP:
2 point general (at least in this class), is an ODE
we have some conditions (or information) about y (or w) at x = a and x
=b,
Math 475B - 8 March 11
Page 1 of 5 - Mar 8, 2011
Hw 3
A) guring out dG/dz in terms of sensitivity mx from hw 2
B) nding the path using newtons method
Math 475B - 8 March 11
Page 2 of 5 - Mar 8, 2011
Math 475B - 8 March 11
Page 3 of 5 - Mar 8, 2011
Math 47
Math 475B - 12 April 11
Page 1 of 6 - Apr 12, 2011
Presentations
How do you plan a presentation?
Know your audience (know what they already are familiar with)
and choose the right level based on what they know and are likely
to be interested in (build on
Math 475B - 5 April 11
Page 1 of 6 - Apr 5, 2011
Draft due 4/26, revised due 5/3
Exam 2 due 4/21
HW 5 due 4/14
Schedule:
PDEs
Method of lines: for evolution equations (parabolic and hyperbolic)
How discretization interacts with time steps
How can you num
Math 475B - 14 April 11
Page 1 of 6 - Apr 14, 2011
Problem 3 on homework 5 was cancelled: going over the problem later
Schedule:
Exam: what's on it and what do we need to be able to do
Problem 2 HW 5
Problem 3 HW 5
Exam is in Math east 145
Boundary value
Math 475B - 7 April 11
Page 1 of 8 - Apr 7, 2011
Math 475B - 7 April 11
Page 2 of 8 - Apr 7, 2011
In a stiff problem, the fast time scale is associated with a rapid decay
and also has slow time scale stuff. Generally not that interested in the
rapid decay
Math 475B - 22 Feb 11
Page 1 of 5 - Feb 22, 2011
Plan:
Go over HW 2 and a sample simpler problem
Do another example of an implicit scheme
Begin shooting method for BVP
In choosing a method:
ode45 because that is the one that you generally try rst
od
Math 475B - 15 Feb 11
Page 1 of 4 - Feb 15, 2011
Figures should:
be labeled, usually titled
be legible, fonts need to be big enough
Every gure should have a point and demonstrate that point clearly
ex: HW 1 7.6 , plot using stiff and non stiff solver:
Math 475B 8 Feb 11
Page 1 of 5 - Feb 8, 2011
Going over Homework 1
7.1: not enough to write:
no long lists of numbers! What information are you trying to convey
with the large lists of numbers, each gure should say something,
don't just put all output.
I
Math 475B - 10 March 11
Page 1 of 7 - Mar 10, 2011
1) How to get a good initial guess for Newton: Homotopy method
2) Tricks for converting more general problems into BVPs in standard
form
For Newtons method, you want to nd a root
If we have an initial gue
Math 475B - 24 March 11
Page 1 of 5 - Mar 24, 2011
Topics:
Finite differences
How to use Matlab built in bvp4c and bvpinit to solve BVPs
HW 4
Math 475B - 24 March 11
Generally want to use symmetric
Discretizations for BVPs
-Because they match the
symmetr
Math 475B - 1 Feb 11
Page 1 of 4 - Feb 1, 2011
Stiffness of ODE IVP's, ie problem 7.6
in problem 7.6, you numerically
solve:
fast decay time scale
slow time scale
Finding exact solution: whenever something is linear, try integrating
factors
What kind of b