8/24/2011
10.1 Taylor Series Method
8/23/2011
Initial Value Problems
ODE like + 9 = with solution
= 1 3 + 2 3
Consider the initial value problem for a first
order differential equation
= ,
=
I
Project I
Calculus of Variation and Active Contours Model
In this project, you are asked to first extend our discussion on the calculus of variation from a
single function to two functions and then yo
Review for midterm examination 1
9/29/2011
1. I.V.P.
2. What is a numerical solution to an I.V.P.?
3. Taylor series method of order 1, 2, .,m with step length
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
11/3/2011
Summary
11/1/2011
Chapter 14. Boundary Value Problem
1. Shooting Method
IDEA: Solve
1
11/3/2011
2. Finite Difference Method
Parabolic Problem: Model equation
1
2
This method is stable if .
A
MAP 4371 Numerical Solutions of Differential Equations
TR 2:00-3:15, Fall, 2011
Dr. Xin Li
Contacting the Professor
Office Hrs:
Classroom:
Office:
E-Mail:
T 9:00-10:00, R 9:00-11:00
MAP 204
MAP 212
xl
Variational Method for Image Analysis
Derivation of the Euler Equations
(11/2010)
We derive the necessary condition for the minimization of an integral
functional of the form:
b
I [y ] :=
F (x, y, y )
11/18/2011
Deriving the Curve Evolution
Equations
A Detailed Computation
Recall 1 and 2
Recall that
For simplicity, we will assume = 1.
1
11/18/2011
Deriving the Equations
Let = (, ) denote the curv
11/22/2011
Level Sets Method
Formulation and Numerical
Implementation
From Curve to Level Sets Method
Curve evolution equation
(, )
=
0, = 0 ( )
= the unit normal of ,
In level set notation:
So
Pl
10/13/2011
15.1 Parabolic Problem
Finite Difference Method
Classification of PDEs
General form of linear second-order PDEs with
two independent variables
auxx buxy cuyy dux euy fu g 0
Linear PDEs: a
Suppose that we are given a dierential equation
x (t) = f (t, x(t), x (t), t [a, b]
with x(a) = x0 , x (a) = x0 . Show that Runge-Kutta method (RK4) can be
applied to this problem in the following for
11/15/2011
Level Sets Method
Introduction
Images
Curves as isolevels of an image, image as a surface
1
11/15/2011
Parametrized Curves
p
N
T
Arc Length, Unit Tengent, and Curvature
Arc length:
=
=
8/31/2011
10.2 Runge-Kutta of Order 4
8/30/2011
Runge-Kutta 2 Recalled
Find ( + ) from () for ODE
= ,
RK2 is given by
1
+ = + 1 + 2
2
1 = ,
2 = + , + 1
It is of order 2 with accuracy O(3 )
1
8/
9/15/2011
11.2 Higher Order ODEs
Transforming into a system of 1st
order ODEs
Higher Order Reviewed
Heres an th order ODE (with I.V.s)
Set
Then, we obtain a system of Odes of the 1st
order:
1
9/15/
9/27/2011
11.3 Adams-Moulton Method
plus some more on stability, consistency,
and convergence concerns
9/27/2011
AM Method
Consider a system = (, ) with
= 0
Heres the multi-step method called AM
me
10/7/2011
14.1 BVP, 1
Shooting Method
Use our knowledge of IVP to solve BVP
Now, we can solve IVP.
Consider the following IVP of order 2:
= , , , [, ]
= 0 ,
= 0
We can solve this IVP on interval
10/12/2011
14.2 A Discretization Method
10/11/2011
BVP Revisited
= , , , [, ]
= , =
We have looked at how to use IVP to help us
solve BVP using the shooting method
Now, we take a different approa
10/25/2011
Elliptic Problem 2
Finite Element Method
Finite Elements as an Alternative to
Finite Difference Method
The finite-element method has become one of
the major strategies for solving partial
10/24/2011
Hyperbolic Problem
Finite Difference Methods
Advection Equation
Follow this link:
http:/www.cse.illinois.edu/iem/pde/discadvc/
=
Here = ( , ) and = ( , )
Using forward in time and centr
12/1/2011
Summary
After the second midterm
Calculus of Variation
Integral functionals of the form
=
, ,
The input is a function = () from certain
collection of feasible functions. For example,
w