Homework 05. Selected solutions
Exercise 05.1. Let c = c(P, t) denote the concentration (the amount per
unit volume) of a chemical at some point P of a 3D domain at time moment
t > 0. Derive the chemical diffusion equation. While deriving, complete the
fo
Homework 12. NOT COLLECTED
Exercise 12.01. Consider the eigenvalue problem:
2
df
d
m
(ODE)
r
+ r r f (r) = 0,
dr dr
(BC)
f (1) = 0,
1 < r < 2,
f (2) = 0.
Here, m is a non-negative integer.
a) Show that > 0. Hint. Construct an analog of = RQ[f ], present
i
Math 3363 Examination I Solutions
Spring 2014
Please use a pencil and do the problems in the order in which they are listed.
1. A rod of length L (units of length), insulated except perhaps at its ends, lies along
the :r-axis with its left end at coordina
Mathematics 3363
Homework VIII Due November 11
Fall 2010
Instructions. Do the problems in the order in which they are listed, write on only one
side of each page of your paper, and staple your pages. Papers are due at the beginning of
class. No late homew
Homework 11. SELECTED SOLUTIONS
Exercise 11.02. Find the differential problem (, PDE, BC, IC) if its
solution is presented in the following series form:
a)
u(x, y, t) =
X
X
m=1 n=1
Amn sin(2mx) sin(ny) expcfw_ 4m2 + n2 t;
b)
u(x, y, t) =
X
X
Amn cos(2mx
Homework 10. Due Monday, April 19
Exercise 10.01. Consider the Sturm-Liouville eigenvalue problem with
Neumann-Dirichlet (ND) choice of the boundary conditions
d2
+ (x) = 0, 0 < x < ,
dx2
d
(0) = 0, () = 0 .
dx
Using your memory, notes, solutions to the
Mathe tics 3363 Examination III
Summer 2014
Instructions. Please use a pencil and do the problems in the order in which they are listed.
1. Find the solution to
2
$99$55, t) gx1:(a:,t)+etsin27rx fort>0and0<$_<_1,
u(0,t) = 0 for t 2 0,
u(1,t) = 0 for t Z O
I'Y 5+ I;
" Math 3363 Examination II
UMQT/Vama; /6(/3U?A) )L
i Spring 2013
Please use a pencil and do the problems in the order in which they are listed.
You may use the following information Without derivation.
- A proper listing of eigenvalues and eige
Math 3363 Examination III Solutions
Spring 2014
Please use a pencil and do the problems in the order in which they are listed.
No books, notes, calculators, smart phones, or other electronics.
1. Find a proper listing of eigenvalues and eigenfunctions for
An Equilibrium Temperature Distrubution
Problem
PROBLEM: Find the value of for which the following problem has an equilibrium
temperature distribution.
w
2w
(x, t) + x for t 0 and 0 x L,
(x, t) =
t
x2
w(x, 0) = f (x) for 0 x L,
w
w
(0, t) = 1, and
(L, t)
Homework 04.
Due Friday, Feb. 19
Exercise 04.1. In class, I gave the definition of the dot product. Let me
repeat. Let V be a vector space. Binary operator v, w V v w R is a
dot product if it is:
Symmetric: v w = w v;
Linear: (
v1 + v2 ) w = v1 w + v2 w
Homework 01. Selected solutions
Exercise 01.1. In the lectures, I derived the mathematical model of the
temperature distribution in 1D metal rod. By analogy, try to derive the
chemical diffusion equation step by step (see pages 9-10 in the book for
refere
REGULAR TWO-POINT BOUNDARY VALUE PROBLEMS
PHILIP W.WALKER
Suppose that a and b are real numbers with a < b, each of p, q, and w is a continuous
real valued function with domain [a, b], the function p has a continuous first derivative, each of p(x) and w(x
Bonus Problems. SOLUTIONS
GLOBAL FORMULATION. Express the 3D Laplace operator in Cartesian, Cylindrical, and Spherical systems. The assignment should be done in
the common framework by solving Problems 1823 from the book, Section
1.5.
Orthogonal curviline
Math 3363 Examination 11 Solutions
Summer 2014
Please use a pencil and do the problems in the order in which they are listed.
No books, notes, calculators, smart phones, or other electronics.
You may use the following information Without derivation.
o A p
Homework 06. SOLUTIONS
Exercise 06.1 Consider the following ODE:
dG
d
r
= G(r), 0 < r < a .
r
dr
dr
a) Where does this ODE come from?
b) Let < 0. Verify (i.e. PLUG and CHECK) that the functions
G1 (r) = cos
ln(r)
and G2 (r) = sin
ln(r)
are two linearly
EMCF 05
Due 1/30 at 11.59 pm
Log into CourseWare at http:/www.casa.uh.edu
and access the answer sheet by clicking on the EMCF tab.
NOTE: On all problems, choice F is None of the above.
For 1 9, which pattern does the integral fit:
up +1
+ C, p 1
p +1
A.
p
Instructions. Please write on only one side of each page of your paper, use a pencil, and
do the probIems in the order in which they are listed.
w" 1 Solve the heat diffusion problem
agar : @(mytx
u(0,t) r 5,
u(1,t) :- 10, and
143,0) :2 5+5x+sin37rzf0rt20
Mathematics 3363
Review for Examination II
Spring 2010
1. Suppose that each of L and H is a positive number. Derive the solution to
2u
2u
(x,
y)
+
(x, y) = 0 for 0 x L and 0 y H,
x2
y 2
u
u
(0, y) =
(L, y) = 0 for 0 y H,
x
x
u
(x, H) = 0, and u(x, 0) = f(
TWO-POINT BOUNDARY VALUE PROBLEMS
SPRING 2010
PHILIP W.WALKER
Suppose that a and b are real numbers with a < b, each of p, q, and w is a continuous
real valued function with domain [a, b], the function p has a continuous first derivative, each of p(x) and