CAAM 336
·
DIFFERENTIAL EQUATIONS
Problem Set 6
Posted Wednesday 29 September 2010. Due Wednesday 6 October 2010, 5pm.
General advice: You may compute any integrals you encounter using symbolic mathematics tools such as
WolframAlpha, Mathematica, or the Symbolic Math Toolbox in MATLAB.
1. [60 points]
Solve the following boundary value problems using the spectral method.
For each problem, (i) write down the expansions of the right hand side functions as linear combinations
of the eigenfunctions; (ii) write down the sum for the solution
u
obtained from the spectral method;
and (iii) produce a plot showing the sum of the ﬁrst twenty terms in the series for
u
.
For parts (c)–(e) you may use the eigenvalues and eigenfunctions computed in Problem 1(d) of Problem
Set 5, and the results of Section 5.2.3 of the text.
(a)

d
2
u
dx
2
(
x
) =
e
x
,
u
(0) = 1,
u
(1) = 0.
(b)

d
2
u
dx
2
(
x
)

10
u
(
x
) =
e
x
,
u
(0) = 0,
u
(1) = 0.
(c)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Tompson
 Orthogonal matrix, Inner product space, spectral method, symbolic mathematics tools

Click to edit the document details