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
= ,
=
It is understood that is a function of :
= (, )
1
8/24
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 you are asked to apply your results to derive the
partial
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.
16.
17.
18.
19.
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21.
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23.
24.
25.
26.
Eulers met
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 .
Another discretization (backward difference on the right
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
xli@math.ucf.edu
Course Description
This course teaches h
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 )dx, y (a) = y0 y (b) = y1
a
Recall that F is a given fu
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 curve (in ) that evolve with time (an artificial
parameter
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
Plug in the curve evolution equation
1
11/22/2011
Curve E
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, b, c,.,g = f(x,y) only
b 2 4 ac 0 , Hyperbolic (2 rea
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 form:
h
x(t + h) = x(t) + hx (t) + (K1 + K2 + K3 ),
6
1
x
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:
=
= 0
Unit tangent vector (verify:
)
Curvature tensor:
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
8/31/2011
The RK4 Algorithm
There are several versions b
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
9/15/2011
Or, in Vector Notation
We can write the previous
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
method:
Initial with the initial value
Run Runge-Kutta
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 [, ] with a
step length =
by RK4, say.
Now, consider
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 approach: starting
from the boundary values and try to fill i
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
differential equations.
As an illustration, we develop
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 central in space:
1
, + ,
Math Modeling II (Fall 2009, Xin
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,
we may require = , =
, 2 [, ]
1
12/1/2011
Eulers Equatio