HW#6 Due: 3/17/09
EGM6341
Spring 2009
From textbook by Atkinson pp185-194 1. 2. 3. #28a, #35 (read p. 171 of the textbook, as well as notes, for "not-a-knot" condition;) (Least square fit) In many applications, one wishes to correlate a dependent

Solution to Homework #1- EGM6341
1. (a) Assume (x) is continuous on a x b, and consider the average 1 n S = f (x j ) n j =1 with all points xj in the interval [a, b]. Show that S = f ( ) for some in [a, b]. Soln: Because (x) is continuous in the

Solution to
HW#7
EGM6341 Spring 2009
pp. 323-329 #1 Write a program to evaluate I =
b a
f ( x )dx using the trapezoidal rule with n
subdivisions, calling the result In. Use the program to calculate the following integrals with n=2,4,8,16,.,512 1

HW#2
EGM6341
Spring 2014
From the textbook by Atkinson: pp.43-50:
21.
For the following numbers xA and xT, how many significant digits are there in xA
with respect to xT?
a)
xA = 451.023,
xT = 451.01
b)
xA = -0.045113,
xT = -0.04518
c)
xA = 23.4213,
xT =

Solution to Homework #2 21.
EGM6341
For the following numbers xA and xT, how many significant digits are there in xA with respect to xT? a) xA = 451.023, xT = 451.01 b) xA =-0.045113, xT = -0.04518 c) xA = 23.4213, xT = 23.4604 Soln. a) | xA - xT

HW#2 EGM6341 Spring 2009 Due 1/27/09 From the textbook by Atkinson: pp.43-50 : 21. For the following numbers xA and xT, how many significant digits are there in xA with respect to xT? a) xA = 451.023, xT = 451.01 b) xA = -0.045113, xT = -0.04518 c) x

Solution to HW#6 EGM6341 Spring 2009 1. #28a. Find a polynomial p(x) of degree 2 that satisfies p(x0)= y0, p (x0)=y 0, p (x1)=y1 Given a formula in the form of p(x) = y0l0(x) + y0 l1(x) + y1 l2(x). Solution: For the suggested p(x), it is seen that p

HW#1 Note: i) ii) iii)
EGM6341
Spring 2009
iv)
This set of HW comes from diverse aspects in calculus and math analyses. It is intended to serve as a review and to bring you up to speed in mathematical analysis. Series expansion in various forms a

HW#1
Note:
i)
ii)
iii)
iv)
v)
EGM6341
Spring 2014
This set of HW comes from diverse aspects in calculus and math analyses.
It is intended to serve as a review and to bring you up to speed in
mathematical analysis. Series expansion in various forms and acc

I.
Introduction
1.1 Why Numerical Methods?
Example 1.
Steady state heat conduction
y
T/y=0 (insulated)
D
C
Governing equation:
2
Heat flow
T=T 1
T= 0
T/x
=0
How do we determine the heat flow
from wall AB to wall AD?
A
Possible solutions:
B x
1. Experimen

Solution to HW#9
EGM6341
Find linear and quadratic least-squares approximations to f ( x) e x on [-1,1]
using monomial basis, i.e., 1, x, x2 ,., xn . Find the error (relative to the exact
function value) of these two approximations for x=0.5.
Solution:
1.

Solution to
HW#10
EGM6341
pp. 323-329
Problem #1. Write a program to evaluate I
b
a
f ( x )dx using the trapezoidal rule
with n subdivisions, calling the result In. Use the program to calculate the following
integrals with n=2,4,8,16,512
1
a)
I exp( x 2

HW#12
EGM6341 Numerical Method
1. Solve the following BVP using finite difference method
y" - (1 x/4) y = x
y(0) = 1, y(2) = 5.
i) Set up the matrix for yi using x=0.5.
ii) Solve yi for i=2 to 4.
Solution:
bi = 0,
ci = -(1- xi /4), di = xi
Choose h=0.5 (w

Solution to HW#11
Problem #6
Write a computer program to solve y= f(x,y), y(x0)=y0, using Eulers method. Write it to be
used with an arbitrary f, stepsize h, and interval [x0,b]. Using the program, solve y=x2 y,
y(0) = 1, for 0x4, with stepsizes of h = 0.

HW#9
EGM6341
1.
#4, p. 239 (Atkinsons book)
2.
#11, p.241 (Atkinsons book)
3.
#13, p.241 (Atkinsons book)
4. Find linear and quadratic least-squares approximations to f ( x) e x on [-1,1] using
monomial basis, i.e., 1, x, x2 ,., xn . Find the error (relat

CHAPTER VII ODE'SINITIAL VALUE PROBLEMS
Objectives: Develop various accurate & efficient methods for solving
ODEs such as
y' = f(x, y)
7.1
(1)
Taylor series expansion for ODE
Example:
Consider the first order ODE:
y' = -2x - y
(= f(x, y) )
with y(0) = -

CHAPTER V APPROXIMATION OF FUNCTIONS
Objective:
f(x) is given in a very complicated form and is difficult to use,
want to approximate f(x) by a simple polynomial in an interval.
Can we do it?
5.1
How do we do it accurately?
Weierstrass Theoremyes we can!

ChapterIX FiniteDifferenceMethodsforPDEs
9.1 EffectsofConvection,DiffusionandDispersiononTransport
LinearKDVBurgersEquation
ModelEqn.fortransport(ofwave,mass,momentum,energy,)
u
u
2u
3u
+ c = 2 + D 3
t
x
x
x
(1)
c=convection(oradvection)speed
=diffu

Turbulence Lecture 33 Now multiply 2 by y n and integrate w.r.t y between 0 & , integrate by parts a couple of times. Final Result
m +1 m U m +1 y nU + 2 1 m+ 2 n U U U n -1 - U y 1 - dy + y 1 - U i x 1 - U n + 1 x m +1 0 U U 0 0 m nU +1 U m + 1 x
dy dy

Turbulence Lecture 34 Power Law Profiles
y Simple: = U ( x ) ( x ) U
1 n
Where n = n ( x ) This is ok for U behavior at y = 0, y =
But
U 1 U n -1 = y 1 y n n
1
u So that y
y =
U 1 n = 1 0 n n
1
Where typically 7 < n < 10 in T.B.L.
u y =
y =0
U 1 1 = !

HW#3
EGM6341
Spring 2014
The following problems are from the Text book.
pp.117-127:
#2
a)
b)
c)
d)
Write a program implementing the algorithm Bisect given in Section 2.1. Use the
program to calculate the real roots of the following equations. Use an error

EGM 6341
Review Questions/Topics for the Mid-Term Exam
CHAPTER 1
How do you determine the number of significant digits in an approximated (rounded or
chopped) value of a variable compared to its exact value?
What is Taylor series expansion in 1-D and 2-D?