Advanced Applied Mathematics
Laplace equation in spherical coordinates
and the heat equation
Math 6171
1. Find the steady-state temperature distribution inside a sphere of radius 1
(i.e. solve the Laplace equation 2 u = 0) when the surface temperatures
f
Advanced Applied Mathematics I
Fourier series
Math 6171
First Name
November 12, 2015
Last Name
to be due at the beginning of class
You need to show your work. Answers without explanation will not receive credit.
1. Find the norm of the following functions
Advanced Applied Mathematics I
Fourier sine and cosine series
Math 6171
First Name
November 19, 2015
Last Name
to be due at the beginning of class
You need to show your work. Answers without explanation will not receive credit.
1. Sketch the graph of the
Advanced Applied Mathematics I
Math 6171
Bessel functions
First Name
October 22, 2015
Last Name
to be due at the beginning of class
You need to show your work. Answers without explanation will not receive credit.
1. Using the indicated substitutions, find
Advanced Applied Mathematics I
Math 6171
Frobenius Method
First Name
Last Name
October 15, 2015
to be due at the beginning of class
You need to show your work. Answers without explanation will not receive credit.
1. For each of the following differential
Advanced Applied Mathematics
Practice Problems for Test 3
Math 6171
1
1. Find the Fourier series of the function f (x) = x + x2 on the interval
4
[, ]. Sketch the graph of the function to which the Fourier series of
f converges on the interval [3, 3].
2.
Advanced Applied Mathematics
Practice Problems for Test 2
Math 6171
1. power series method
Find the power series solution of the differential equation y 00 + 3y 0 + 2y = 0 at the point
x = 0.
2. regular and singular points
Classify the point x = 0 of the
Advanced Applied Mathematics
Practice Problems for Test 1
Math 6171
1. Find the length of the curve ~
r (t) = (8t, 4t2 , 4 ln t) between the points (8, 4, 0) and (32, 64, 4 ln(4).
2. Find the directional derivative of the function f (x, y, z) = y cos2 z (
f ( t ) = L -1 cfw_F ( s )
1.
1
3.
t n , n = 1, 2,3,K
5.
1
s
n!
s n +1
p
t
3
2
7.
sin ( at )
9.
t sin ( at )
11.
Table of Laplace Transforms
F ( s ) = L cfw_ f ( t )
sin ( at ) - at cos ( at )
2s
a
2
s + a2
2as
(s
2
+ a2 )
2a 3
2 2
2
cos ( at ) - at sin (
HEAT CONDUCTION THROUGH
FINS
Prabal Talukdar
Associate Professor
Department of Mechanical Engineering
IIT Delhi
E-mail: prabal@mech.iitd.ac.in
p
Mech/IITD
Introduction
&
Q conv = hA s (Ts T )
There are two ways to increase the rate of heat transfer
to in
Conduction Heat Transfer Notes for MECH 7210
Daniel W. Mackowski Mechanical Engineering Department Auburn University
2
Preface
The Notes on Conduction Heat Transfer are, as the name suggests, a compilation of lecture notes put together over 10 years of te
Heat Conduction and the Heat Equation
If you cant take the heat, dont tickle the dragon.
Heat Transfer
Note: Energy is the conserved quantity
Conduction:
Heat transfer due to molecular activity. Energy is transferred from more
energetic to less energetic
Part G-3: Solved Problems
Part G-3: Solved Problems
MPE 635: Electronics Cooling
1
Part G-3: Solved Problems
1. A square silicon chip (k = 150 W/m. K) is of width w =5 mm on a side and of thickness t = 1
mm. The chip is mounted in a substrate such that it
PROBLEM 1.1
KNOWN: Heat rate, q, through one-dimensional wall of area A, thickness L, thermal
conductivity k and inner temperature, T1.
FIND: The outer temperature of the wall, T2.
SCHEMATIC:
ASSUMPTIONS: (1) One-dimensional conduction in the x-direction,
Heat and Mass Transfer
Syllabus
HEAT AND MASS TRANSFER
Module 1: Introduction (2)
Units, definitions, Basic modes of Heat transfer, Thermal conductivity for various types of
materials, convection heat transfer co-efficient, Stefan Boltzman's law of Therma