Heat Problem
c : 1.04
In[55]:=
Out[55]=
In[41]:=
0.056
10.6 ;
c
1.75202
2
10
Integrate
4
0.8 Abs x
1.62114 2. Sin 1.5708 n
5
Sin n Pi x
10 , x, 0, 10
1. Sin 3.14159 n
Out[41]=
n2
Initial Distribution
In[61]:=
Plot
4
0.8 Abs x
5
, x, 0, 10
4
3
Out[61]= 2
1
In[12]:=
f x_ : k 1
Cos Pi x
;k
0.01;
In[13]:=
Plot3D 1 2 f x t
f x t , x, 0, 1 ,
t, 0, 10 , AxesLabel Automatic, Mesh None
0.0
x
0.5
1.0
0.020
0.015
Out[13]=
0.010
0.005
0.000
10
5
t
0
In[32]:=
c : 1.04
0.056
In[33]:=
p x_, t_, k_ : Sum
16
1 ^n
n Pi
32
n
In[48]:=
? BesselJ
BesselJ n, z gives the Bessel function of the first kind Jn z .
In[170]:=
Plot BesselJ 0, x , BesselJ 1, x ,
x, 0, 13 , PlotLegends Placed "Expressions", Below
1.0
0.8
0.6
0.4
Out[170]=
0.2
2
4
6
8
0.2
0.4
J0 x
J1 x
10
12
2
BesselJ.nb
I
Math 315-01
Exam 2
Spring 2010
Name:
ID:
Date:
Directions: Show all work. No work equals no credit. You may use a basic scientic calculator, but no graphing/symbolic/cell phone calculators. Use proper
notation.
(8 pts) 1. Consider the initial value proble
Math 440/540
Exam 2
Fall 2013
Name:
ID:
Date:
Directions: Please show all work unless specied otherwise. You may use a basic
scientic calculator, but no graphing/cell phone calculators. Use proper notation.
(15 pts) 1. Find the Fourier series of the funct
1 Math 440/540 Exam 1 Review Sheet
1
1
Math 440/540 Exam 1 Review Sheet
The exam covers sections 5.1, 5.3, 5.4, and 5.5. No graphing/symbolic/cell
phone calculators allowed and no notes are allowed. You may bring a simple
scientic calculator. Dont be late
Math 440/540
Final Exam
Fall 2013
Name:
ID:
Date:
Directions: Please show all work unless specied otherwise. You may use a basic
scientic calculator, but no graphing/cell phone calculators. Use proper notation.
(15 pts) 1. Use the Frobenius Method to nd t
In[44]:=
f x_ : Integrate
1
In[45]:=
Cos Pi w
w Sin w x , w, 0, Infinity
Plot f x , x, 0, 2 Pi
1.5
1.0
Out[45]=
0.5
1
In[46]:=
2
Plot Piecewise
3
Pi
2, x
4
5
Pi , 0, x
6
Pi
, x, 0, 2 Pi
1.5
1.0
Out[46]=
0.5
1
2
In[47]:=
g x_ : Integrate
In[48]:=
Plot g x
Mass-Spring System with Aging Spring
Definitions
a: 1
20; k : 100; m : 1; y0 : 1; v0 : 1
f t_ : 2
a
Sqrt k
m
E^
at
2
f t
400
t 40
Results
Bessel function of first kind only.
s t_ : y0
s 0
1
N
1.
BesselJ 0, 2
a
Sqrt k
m
BesselJ 0, f t
2
Bessel-Spring.nb
Pl
Example
In[328]:=
In[329]:=
f x_, n_ : 1 8
1 ^k 1
Sum
2 k ^ 2 Pi ^ 2 Cos 2 k Pi x
1 ^k
2 k Pi Sin 2 k Pi x , k, 1, n
f x, 1
1
Cos 2 x
8
2
Sin 2 x
Out[329]=
In[334]:=
2
Plot f x, 100 , x,
1
2, 1
2
0.5
0.4
0.3
Out[334]=
0.2
0.1
0.4
0.2
0.2
0.4
2
Fourier-2.n
Fourier-Bessel Series
Fourier-Bessel Series for f(x)=(1+x)^2 on the interval 0<x<2 using J_0(x) as the
basis.
In[219]:=
g x_ :
In[220]:=
r: 2
In[221]:=
a m_ : 2
r ^ 2 BesselJ 1, N BesselJZero 0, m
^2
NIntegrate x g x
BesselJ 0, N BesselJZero 0, m
r x , x,
Mass-Spring System with Aging Spring
Definitions
In[202]:=
a: 1
20; k : 100; m : 1; y0 : 1; v0 : 1
In[10]:=
f t_ : 2
In[11]:=
f t
Out[11]=
400
a
Sqrt k
m
E^
at
2
t 40
All Mathematica Solution
Using y0 and v0 from above. Note that Mathematica can solve the
BK A Z 1 2 \ - z ZWMZWoszwl
)7-3 @ jglw) = J}; {0X res [wank > J;[smwx+ ices/mil: F{._.____.J:_____ .
A (L 1. I
® 99) = Er C" 5(wa 7 sRWH £iS/M) ~55 {Raw} /0
2 S'Mw 2,05 129.3(4) 2 wlfliaw +214 (4w ~2yiw
- + - a » __________._____
" n w WI W3 IT W3
w
In[11]:=
? LegendreP
LegendreP n, x gives the Legendre polynomial Pn x .
LegendreP n, m, x gives the associated Legendre polynomial Pm x .
n
In[27]:=
Plot Evaluate Table LegendreP n, x , n, 0, 5 ,
x, 1, 1 , PlotLegends Placed "Expressions", Below
1.0
0.5
Math 440/540 SL Conversion
For #13 in the HW for next week you will need to adjust the given ODE so it is in SL
form. The problem I gave with the term xy + y collapses into (xy ) . So not much to do,
just product rule backwards.
The other one I gave had y
1 Power Series
1
1
Power Series
There seems to be a few questions regarding power series. I will try to address
them.
1.1
Denition
an xn = a0 + a1 x + a2 x2 + a3 x3 +
n=0
1.2
Reindex
Start the sum at n = 2 shift nth term down by two.
an2 xn2 = a0 + a1 x
