Math 442
Takehome Exam
Due March 2
Texas A & M University
Spring 2011
1. Consider the lake purication model:
r
x (t) = (1 + sin(2t)x(t),
v
where x(t) is the concentration of contaminant (gr/m3 ), v is the volume of the lake (m3 ), r is
the mean outow, t
Dimensional analysis problems
Math 647.600
1. Why do stringed musical instruments have strengths of dierent lengths and thicknesses? Assume that the fundamental frequency of vibration of a string depends on its length l, mass per unit length , and tension
Dimensional analysis problems Math 442.500 1. Why do stringed musical instruments have strengths of dierent lengths and thicknesses? Assume that the fundamental frequency of vibration of a string depends on its length l, mass per unit length , and tension
Math 442, Quiz # 6
Name & ID number:
(Solutions)
Texas A & M University
Spring 2011
[15 points] 1. The Poissons equation related the velocity w of uid in an elliptical pipe is
given by ( 2 w(x, y )/x2 + 2 w(x, y )/y 2 ) = (p1 p2 )/L. What is the solution
Math 442, Quiz # 1 (solutions)
Name & ID number:
Texas A & M University
Spring 2011
[30 points] 1. What are the main assumptions of the lake purication model discussed in section
1.1 of the textbook?
1.
2.
3.
4.
The lake has a constant volume.
The lake is
Math 442, Quiz # 2 (solutions)
Name & ID number:
Texas A & M University
Spring 2011
[20 points] 1. What is the essential perquisite of using metered models? Why such models are
called memoryless?
Essential Prequisite: The system being modelled must be qua
MATLAB for M151B
c 2008 Peter Howard
1
MATLAB for M151B
P. Howard
Fall 2008
Contents
1 Introduction
1.1 The Origin of MATLAB . . . . . . . . . . . .
1.2 Our Course Goal . . . . . . . . . . . . . . . .
1.3 Starting MATLAB at Texas A&M University
1.4 The MA
M442, Fall 2016 Practice Problems for the Midterm
The midterm for M442 will be Wednesday, Oct. 26, 79 p.m., in Blocker 122. The exam will
consist of two parts: Part 1 will not require MATLAB, while Part 2 will require MATLAB.
Students will turn in Part 1
M442, Fall 2016, Assignment 4 Solutions
1. [10 pts] An object is shot straight upward with initial velocity v. Ignoring air resistance,
and assuming gravity is the only force acting on the object, use dimensional analysis to
determine the general form (i.
M442 Assignment 5 Solutions
1. [5 pts] For the system of differential equations
x = 1 + ax2 by 2
y = by + axy,
suppose the point (xe , ye ) = (1, 2) is known to be an equilibrium point. Use this to find
values for the parameters a and b.
Solution.
This eq
M442, Fall 2016, Assignment 3 Solutions
1. [10 pts] Suppose we have data cfw_(tk , yk )N
k=1 for which the independent variables are equally
spaced, with
h = tk+1 tk
for k = 1, 2, . . . , N 1. Show that
y 0 (tk ) =
y(tk + h) y(tk h)
+ O(h2 ),
2h
for k = 2
Math 442, Quiz # 5
Name & ID number:
(Solutions)
Texas A & M University
Spring 2011
[15 points] 1. What is the process of validating a model? When can we accept a model?
First we propose a test (e.g. whether the model predictions match with the recorded d
Math 442, Quiz # 4
Name & ID number:
(Solutions)
Texas A & M University
Spring 2011
[30 points] 1. (a) What are the main benets of nondimensionalization? (b) Nondimensionalize
the plant growth model (1.15).
a) The main benets are: (1) the relative strengt
Notes for TUT course MAT51316 Robert Pich e 2.9.2007 1 Transport Equation Initial Value Problem
how to derive the one dimensional transport equation how to solve initial value problems for this equation using the method of characteristics how to
Notes for TUT course MAT51316 Robert Pich e 4.9.2007 3 Maximum Principle for 1D Diffusion Equation
Theorem: If ut = kuxx in (0, l) [0, T ] then u achieves its maximum in R = [0, l] [0, T ] at t = 0 or at x = 0 or at x = l (and possibly elsewhere
Notes for TUT course MAT51316 Robert Pich e 10.9.2007 4 Plot Solution of 1D Diffusion IVP
The solution of the one dimensional diffusion equation ut = kuxx with initial condition u(x, 0) = (x) is given by the convolution integral
u(x, t) =

S(x
Notes for TUT course MAT51316 Robert Pich e 4.9.2007 2 Wave Equation Models
vibrating string vibrating membrane
How to derive the PDE for
2.1
Vibrating String
Consider the motion of a thin string moving in the xz plane. Assume that points of t
1. First, we plot both curves on the same set of axes. A little experimentation gives a reasonable range. I'm defining these as functions so that I don't have to type these out again when I want to plug in later. You could write them as expressions as wel
Problem 1: We will want to find and so that the partials to be zero:
is minimized. Set
Collecting terms and recalling that
, this gives two equations in two unknowns:
From Cramer's rule, we get
and
Problem 2:
I'll call the month array
and the temperature
Math 442
Homework # 1
Solutions
Texas A & M University
Spring 2011
1. You are a professional modeler at Texas Department of State Health Services. You have
been assigned to study an inuenza epidemic which occurred in a boarding school in the north
of Dall
Math 442
Homework # 2
Due February 23
Texas A & M University
Spring 2011
1. Consider the preypredator model
dx(t)/dt = x(t)(a1 b1 y (t),
dy (t)/dt = y (t)(a2 + b2 x(t).
a ) [10 points] In terms of conversion of energy from one source to another, is it po
Math 442
Homework # 4
Due April 4
Texas A & M University
Spring 2011
1. The simple pendulum consist of a particle P of mass m suspended from a xed point O by
a light string of length a, which is allowed swinging in a vertical plane. If there is no frictio
M442 Assignment 2 Solutions
1. [5 pts] For a pendulum under the influence of linearly modeled air resistance, the angle
y(t) solves the equation
g
y (t) = sin y by ,
l
where b is a coefficient of air resistance. Taking l = 1, g = 9.81, and b = .1, solve t