# Study Guide Test #2

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Study Guide
418/518
Test #2
Study Guide
Test #2
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Problem #1:
Use the method of separation of variables to solve the wave equation
utt  c2 uxx
on the interval 0 x L subject to boundary conditions of the first or second kinds at
x  0 and x  L and given initial conditions
u(x,0)  f (x), ut (x,0)  g(x) (0  x  L) .
Read Section 4.4
Review sections 2.3, 2.4.1, 2.4.2 and problem #3 of Test #1. Review problems 2.3.2
(a)-(f), and (2.3.3) (a)-(d) on page 39 of text. Understand and memorize the table on
page 69. Review problems 2.4.1 and 2.4.2 on pages 69-70 of text.
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Problem #2:
Use the Method of Eigenfunction Expansion to solve the one dimensional heat equation
ut  kuxx  q( x, t )
(0  x  L, k  constant, k  0)
ut  uxx  q(x,t) (0  x  L,t  0) subject to an initial condition u(x,0)  f (x) (0  x  L) ,
and homogeneous boundary conditions of the first or second kind.
Read Method of Eigenfunction Expansion on pages 122-124. Do problems (3.4.8),
(3.4.9), (3.4.10), (3.4.11), (3.4.12) on page 126.
Read section 8.3 pp. 354-358, especially the Example on page 357. Do problem 8.3.1
(a) and (f). Also do problems (8.3.2) , (8.3.6) and (8.4.4) on page 364.
Read supplement I.
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Problem #3:
(a)-Find the equilibrium solution ue (x) of the following initial-boundary value problem:
ut  kuxx (0  x  L,t  0, k
0) , u(x,0)  f (x) (0  x  L) ,
u(0,t)  A, u(L,t)  B or u(0,t)  A, ux (L,t)  B or ux (0,t)  A, u(L,t)  B
where A and B are constants.
(b) -Using the result of part (a) find the partial differential equation, the initial condition,
and the boundary conditions satisfied by v(x,t)  u(x,t)  ue (x) where u(x,t) is the
solution of the initial-boundary value problem stated in part (a).
(c)- Find the ordinary differential equations, and initial conditons sautisfied by the
functions bn (t) (n  1,2,3,....) where v(x,t) 
n 
 b (t) (x)
n
n
is the solution of the initial-
n1
boundary value problem derived in part (b) found by applying the Method Of
Eigenfunction Expansion.
Read Method of Eigenfunction Expansion on pages 122-124. Read sections (8.1)
and (8.2). Do problem (8.2.1) on page 352.
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418/518
Study Guide
Test #2
418/518
Study Guide
Test #2
Problem #4:
Let u(x,t) solution of the following initial-boundary value problem:
u(0,t)  A(t), u(L,t)  B(t)
or
u(0,t)  A(t), ux ( L,t)  B(t)
or
ux (0,t)  A(t), u( L,t)  B(t)
(a)-Construct a function r(x,t) that satisfies the same boundary conditions as u(x,t) .
Hint: Set r(x,t)   (t)   (t)x , and find  (t) and  (t) .
(b) -Using the result of part (a) find the partial differential equation, the initial condition,
and the boundary conditions satisfied by v(x,t)  u(x,t)  r(x,t) where u(x,t) is the
solution of the initial-boundary value problem stated in part (a).
(c)- Find the ordinary differential equations, and initial conditons satisfied by the
functions bn (t) (n  1,2,3,....) where v(x,t) 
n 
 b (t) (x)
n
n
is the solution of the initial-
n1
boundary value problem derived in part (b) found by using the Method of Eigenfunction
Expansion. Read Method of Eigenfunction Expansion on pages 122-124. Read
sections (8.1) and (8.2). Do problem (8.2.1) on page 352.
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Problem #5:
Given the graph of a piecewise smooth function f (x) defined on the interval 0  x  L :
(a)- Sketch the graph of f e (x) the even extension of f (x) on the interval L  x  L .
(b)-Sketch the graph of f p (x) the periodic extension of f e (x) on the interval 3L  x  3L .
e
 n x 
be the Fourier cosine series of f (x) . To what value does this
L 
n0
Fourier series converge at x  0, x  L, x  L, x  3L, x  3L, and at each point in the
interval 0 x L where f (x) has a jump discontinuity?
Given the graph of a piecewise smooth function f (x) defined on the interval 0  x  L :
(d)- Sketch the graph of f o (x) the odd extension of f (x) on the interval L  x  L .

(c)-Let
 A cos 
n
(e)-Sketch the graph of f p (x) the periodic extension of f o (x) on the interval 3L  x  3L .
o
 n x 
be the Fourier sine series of f (x) . To what value does this
n
L 
n0
Fourier series converge at x  0, x  L, x  L, x  3L, x  3L, and at each point in the
interval 0 x L where f (x) has a jump discontinuity?
Read section (3.3) pp. 96-109 and pp.109-111. Also read (3.3.5) pp. 111-113.
Do problems (3.3.1)-(3.3.2), and (3.3.4).
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
(f)-Let
 A sin 
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418/518
Study Guide
Test #2
Problem #6:
(a)-Compute the Fourier series of the function
 g(x)  L  x  0
f (x)  
 h(x) 0  x  L
where g(x)  ax  b and h(x)  cx  d .
(b)-Sketch the function to which the Fourier series of part (a) converges on the interval
3L  x  3L . Be careful to indicate the values to which the series converges at
x  3L, x  2L, x  L, x  0, x  L, x  2L, x  3L.
Read section (3.2). Do problems (3.2.1), (3.2.2) on page 95.
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