Partial Differential Equations 3150
Final Exam
Exam Date: Friday, April 25, 2014
Instructions: This exam is timed br 120 nilnutes.
Yu may ho given ext;ra time to complete
lie vxai (10—15 extru nunutes). No calculators, notes,
tables or books. Problems use topics from
the required textbook which were covered in lect;ures.
Details count 3/4, answers count 1/4.
1. (Chapters H1-.H2. Heat Conduction in a Bar)
Considered is the heat conduction problem
Ut
=
u(0,i)
=
u(x,0)
4
A
L.r,
=
20,
—20,
=
f(s),
0
<5
<8,
t > 0,
t > 0,
t>0,
(1)
0 < x < 8.
It represents a laterally insulated uniform bar of length
8 with one end held at 20 Celsius
and the other end held at -20 Celsius, and initial temper
ature f(s).
1(a) [30%] Show the details for finding the the steady-state
temperature u
1 (z).
1(b) [40%] Show the details for the solution u
(x,t) of the ice-pack ends bar problem
2
=
u(0,t)
u(,t)
=
,
1
‘u
0<x<8, t>O,
0,
0,
100
u(x,0)
t>0,
t>0,
(—1sin(2n7rx/8), 0 < x < 8.
1(c) [30%] Superposition implies u(z, t) = u
1 (x, t) + u2(x, t) is a solution of (1) with
j-ui (z,ê+ u2(x, 0). Show the details of an answer
check for the solution u(s, t).
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2. (Chapter 1-13. Fourier Series)
A
,
fut inn sin(:r) hus n Fourier
2(a) f2O j
sat islils
(‘(501w
1+
a
sin(a’)
values
sin(a)
to eoiiipiite
the
ci a,
(I. 1, Theii
cos(b)
ternis with b
011
(1
<
:r
<
ir. The series
(, coH(?cC).
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series rlcfiiwd
I
—
use the trig identity
(siu(a
-
sin(a
b)
—
b))
0.
2(b) [40%] Let q(n) be the Fourier cosine series for the period 2exnai of the
function qo(x)1on 0 < a: 1/2, go(a:) —1 on 1/2 < x 1. Complete the following.
(1)
(2)
(3)
(4)
(5)
A
Graph of Pu with its even extension on al
Graph of g(x) over 3 periods.
Fourier cosine series coefficient formulas.
Numerical values for the coefficients.
Gibbs overshoot graphic on al < 1.
2(c) [20%] Define ho(x)
=
{
—
<
0X<7
1.
and let h(x) be the 4ir odd periodic
ir
extension of ho(x) to the whole real line. Compute the sum h(—5.5w) + 2h(4.5ir) + 3h(7ir).
.—2(d) [10%] Compute a smallest period, for those functions which
(1) fi(x) =3siu(z)+7cos(/r) 1
(2) f
(z) = sin(7rx) + sin(irz/7) + sin(2xx/15)
2
(x) = sin(1/x) cos(v’x))
3
(3) f
1+sin(x)
(4 f()
}‘
2+cos(vx)
f
are
periodic.
P
2_I”
€
.
—
—
10
(5) f(x)=
siri(mirx)
2 )
1+n
2(e) [10%] A complex Fourier series
>
2 of a real function f(x) has to equal its
e
n—oo
Fourier series
ao+acos(nx)+b,sin(n.x).
,
0
Briefly explain how to find the equations relating the sequences {a}
but don’t supply derivation details or display the equations.
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3150 Final Exani S204
3. (CH H4. Finite String: Fourier Series Solution)
3(a) [50% Complete the following for the finite string problem
11 (v t)
u
u(0,t)
u(L, t)
u(x,0)
u(x,0)
(1)
(2)
(3)
(4)
(5)
c
u
2
rr(1:, t),
0,
0,
(1 < Z < L,
=
f(x),
=
g(x),
0 < c < L,
0< z < L.
=
=
t > 0,
t >0,
t > 0,
Separation of variables details.
Product solution boundary value problems.
The product solutions (normal modes).
Superposition.
Series solution u(x,t).
3(b) 125%] Display explicit formulas for the generalized Fourier coeffic
ients which contains
the symbols f(x), g().
3(c) [25%] Evaluate the coefficients when c
1/3, L
=
5,
f(s)
=
0 and g(z)
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313() Fitial Exam S2014
Namc.
4. (Cl-I H4. Rectangular Membrane)
Coiiipletc tlI(’ lollowing for tin’ g(’lwrol nwiiihra t’ pr()l)i(ifl
:z:,y,t)
(u
0
‘u(x, y, t)
=
.—
u(x,y,0)
uj(z,y,0)
undur the
r (‘ r.r(r,y,t) -l ‘u(r,j,t)),
2
0
f(a:,ij),
=
asiuui1ptio1iH
gQr,ii),
a
=
2, b
4, r
=
i., f(:i:, y)
o
0 <:r < a, 0 < y <h, 1 > 0,
oil tlw boundary,
0 < a: < a, 0 < p < b,
(I < i: < a, 0 < p < 1.).
0, q(a’, y)
=
100.
(1) Separation of variables.
(2) Product solution boundary value problems.
(3) Product solutions (the normal modes).
(4) Superposition for v,(z,y,t).
(5) Generalized Fourier coefficient formulas.
(6) Explicit numerical values for the coefficients.
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3150 Fiiial Exam S2014
5. (Cl-I 1-110. Fourier Transform, Heat Equation and Gauss’ Heat Kernel)
5(a) [20%) TIn! ermnn ‘s Fourier transform pan is
A—
f(:v)rturd:r,
=
Assunw [(fr) = 21r(”’ on 0
F(w) of f(:r).
Graded Details: (1)
<
r < 1 and f(:r)
=
f(:r)
F(w)ec1w.
(1 otherwise. Compute the Fourier transform
(2) Integration details, (3) Answer.
Transform integration setup,
2iu’
5(b) [10%) Compute the magnitude of the Fourier transform F(w)
=
The phase 0 of a Fourier transform F(ui) is defined as the solution of the equation
G y 5(c) [10%)
F(w)/)F(w). What is the phase of F(w)
=
5(d) [50%) Gauss’ heat kernel g(x) (k, t
by the equations
e
constants)
Sin,
erf(x)
=
The formula FT[g(x)J
problem
G(w)
=
=
fQr)
=
2
=
(X2
J
e
dz.
2 is assumed. Solve the infinite rod heat conduction
e°
—oo<x<oo,
=
u(x,0)
and the error function erf(x) are defined
f(z),
t>0,
—oo<x<oo,
120 0<x<1,
0 otherwise
Graded Details: (1) Fourier Transform method, (2) Convolution, (3) Heat kernel use, (4)
Error function methods, (5) Final answer, expressed in terms of the error function.
A
5(e) [10%) Compute the limits u(0, 0+) and u(1, 0+), which are the right hand limits of
temperature n(x, t) at t 0 for rod locations x = 0 and x = 1.
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