AE5327/ MAE4301
Introduction to Computational Fluid Dynamics
Homework #3
Due 11/13/14
Given the following model equation:
d
d
+ 0.1
= 0.0
dt
dx
Perform the following tasks:
a) Derive the amplification factor for these schemes:
i. forward Euler with first-

MAE 5331 Fall 2011
Analytical Methods in Engineering
Prof. Albert Y. Tong
Lecture Note on Partial Fractions
Simple Rules on Partial Fractions
i.
If the degree of the numerator of the given function is equal to or greater than that
of the denominator, divi

ME 5331
Analytical Methods in Engineering
Prof. Albert Y. Tong
Ordinary Differential Equations: Part II
Linear Differential Equations of Higher Orders
As a first step, we must look at the idea of linear independence.
Consider the set of functions: u1 (t

ME 5331
Analytical Methods in Engineering
Prof. Albert Y. Tong
Ordinary Differential Equations: Part I
Basic Nomenclature
Consider a general O. D. E. with t as the independent variable, and y the
dependent variable.
a 0 ( y, t )
dny
d n 1 y
dy
a1 ( y, t

Dot and Cross Product
October 31, 2007 - Happy Halloween!
Dot Product (Inner product) Denition: Let a and b be two vectors in Rn, then the dot product of a and b is the scalar a b given by
a b = a1b1 + a2b2 + a3b3 + + anbn
1
Properties of the Dot Product

1
2
Delta functions and Heaviside step functions
Denition of Dirac delta function
The denition of the Dirac delta function is
(x) f (x)dx = f (0),
(1)
i.e. (x) is dened by its action as an operator and is therefore different from ordinary functions. The

LIMIT COMPUTATION: FORMULAS AND TECHNIQUES, INCLUDING
LHOPITALS RULE
MATH 153, SECTION 55 (VIPUL NAIK)
Corresponding material in the book: Sections 11.4, 11.5, 11.6.
What students should already know: Basic limit computations.
What students should denitel

22M:132 Fall 07 J. Simon
Handout 3
More Discussion of Section 17 limit point closed set closure interior boundary
This section introduces several ideas and words (the ve above) that are among the most important and widely used in our course and in many ar

Lagrange multiplier
As an application of the theory of implicit functions, the Lagrange multiplier method is discussed
in this lecture to determine the extremum (either a maximum or a minimum) of a function f(x,y)
subjected to the constrain g(x,y)=0. Lets

Partial fraction
A0 + A1 s + A2 s2 + Am1 1 sm1 1
P(s)
=
+.
(s a1 )m1 (s a2 )m2 (s an )mn
(s a1 )m1
(1)
s+1
a0 + a1 s + a2 s2
b0
=
+
.
3 (s + 2)
3
s
s
s+2
(2)
Example:
In order to determine the unknowns, the denominators can be eliminated as
s + 1 = (a0 +

Laplace transforms
The formulae of the Fourier transform and the inverse Fourier transform are listed here for reference as
1
2
F (x) =
F ( ) =
F ( )ei x d ,
(1)
F (x)ei x dx.
(2)
The Laplace transform can be derived as a special case of the Fourier trans

Preliminary
Recall:
Question: how about the case when l tends to infinity? In this case, f(x) is actually a nonperiodic function over the whole domain < x < . As demonstrated before, a
periodic function can be represented by a Fourier series. If we can tr

MATH3530 Course notes
1
Copyright (C) 2013 Neil Rohan Ramsamooj. Permission is granted to copy, distribute and/or
modify this document under the terms of the GNU Free Documentation License, Version 1.3
or any later version published by the Free Software F

ME 5331
Analytical Methods in Engineering
Prof. Albert Y. Tong
Ordinary Differential Equations: Part III
Solution of Non-homogeneous Equations
Beginning with the
order linear O.D.E.
The Existence and Representation theorem is used to guide our approach.

AE5327/ MAE4301
Introduction to Computational Fluid Dynamics
Homework #2
Due 10/28/14
Given the following model equation:
d 2
0.1 x 2
2
dx
Perform the following tasks:
a) Derive the exact solution to the equation for the boundary conditions
(1.0)=ddx(0.0

Combustion HW 3
1.
Consider the hypothetical reaction
A+BC+D
The reaction as shown is exothermic. Which has the larger activation energy, the exothermic
forward reaction or its backward analog? Explain. (Hint see the Graph from class on activation
energy)

Project Ideas 2015
The project report should include the following sections: introduction/problem
background, methodology, results, conclusions, references, and code listing (if a program
was developed). The report is due by email at or before the end of

Homework Chapter 4
Due 11/01/2016
1. A stoichiometric fuelair mixture flowing in a Bunsen burner forms a well-defined conical flame. The mixture is then made leaner. For the
same flow velocity in the tube, how does the cone angle change for the leaner mix

Combustion Homework 1
DUE 09/15/2016
1. Suppose that methane and air in stoichiometric proportions are
brought into a calorimeter at 500 K. The product composition is
brought to the ambient temperature (298 K) by the cooling water.
The pressure in the cal

AE5327/MAE4301
Introduction to Computational Fluid Dynamics
Homework #1
Due 10/9/14
1. Starting from the dimensional form, derive the non-dimensional form of the
unsteady incompressible Navier-Stokes equations (without gravity & energy
equation). The fina

Combustion Homework 2
1. A combustion test was performed at 20 atm in a hydrogenoxygen system. Analysis of the
combustion products, which were considered to be in equilibrium, revealed the following:
Compound
Mole fraction
H2O
H2
0.493
0.498
O2
O
0
0
H
OH

AE5327/ MAE4301
Introduction to Computational Fluid Dynamics
Homework #4
Due 11/25/2014
Given the following model equation and boundary conditions:
2 2
+
= 1.0
2 x 2 y
(0, y ) = (1, y ) = ( x,1) = ( x,0) = 0
Perform the following tasks:
a) Write a progr

Innite Series
Innite series are among the most powerful and useful tools that youve encountered in your introductory
calculus course. Its easy to get the impression that they are simply a clever exercise in manipulating
limits and in studying convergence,

Trapezoidal Rule Example
Use the Trapezoidal rule to approximate
0
sin x dx
using
1. n = 10 subintervals,
2. n = 20 subintervals, and
3. Richardson extrapolation.
Here a = 0, b = , and f (x) = sin x.
1. n = 10 = h =
ba
=,
n
10
and
xi = a + ih.
i
0
1
2
3
9

TheCrossProduct
ThirdTypeofMultiplyingVectors
CrossProducts
r
r
r
r
u
r
r
r
r
If v = a1 i + b1 j + c1 k and w = a2 i + b2 j + c2 k
are two vectors in space, the cross product
ru
r
v w is defined as the vector
ru
r
r
r
r
v w = (b1c2 b2 c1 )i (a1c2 a2c1 ) j

1
Note that we have used to the Fourier series to solve the ODE of periodic nature before.
Right now, we can also use the Fourier transform, together with the inverse Fourier
transform, to solve an ODE with no periodic nature.
2
3
4

precalculus
algebra quick guide - exponents
product of like bases
xa xb = xa+b
x3 x4 = ( x x x) ( x x x x) = x7
quotient of like bases
xa
= xa
xb
b
, x 6= 0
x5
xxxxx
=
= x3
x2
xx
power to a power
(xa )b = xab
( x 2 ) 3 = ( x 2 ) ( x 2 ) ( x 2 ) = (x x ) (