kma354 partial differential equations applications & …
TRANSCRIPT
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Student ID No:Pages: 6
Questions : 5
UNIVERSITY OF TASMANIA
EXAMINATIONS FOR DEGREES AND DIPLOMAS
November 2010
KMA354 Partial Differential EquationsApplications & Methods 3
First and Only Paper
Examiner: Dr Michael Brideson
Time Allowed: TWO (2) hours.
Instructions:
- Attempt all FIVE (5) questions.
- All questions carry the same number of marks.
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KMA354 Partial Differential Equations 3, 2010 - 2 -
1. (a) Explain how and why the tangent plane can be used to solve a first order
quasilinear partial differentiation equation.
(b) Solve the following Cauchy problem using the Method of Characterisitics.
∂U
∂x+ (1 + y)
∂U
∂y= x ,
U(x, 0) = sin((1 + x)2).
(c) A Cauchy problem is solved using the Method of Charateristics to give
U(x, y). Explain the relationship between contours of U and the character-
stics.
Continued . . .
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KMA354 Partial Differential Equations 3, 2010 - 3 -
2. Consider the nonhomogeneous wave equation
∂2U
∂t2−
∂2U
∂x2= −1 , 0 < x < L, t > 0 ;
with initial and boundary conditions,
BCs : U(0, t) = 0 , t > 0 ;
U(L, t) = 0 , t > 0 ;
ICs : U(x, 0) = 1 +x
2(x− L) , 0 < x < L ;
∂U
∂t(x, 0) = 0 0 < x < L.
Make the substitution U(x, t) = v(x, t) + ψ(x) and show the solution to be
U(x, t) =x
2(x− L) +
4
π
∞∑
n=0
(
1
2n+ 1
)
sin
(
(2n+ 1)π x
L
)
cos
(
(2n+ 1)π t
L
)
.
Continued . . .
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KMA354 Partial Differential Equations 3, 2010 - 4 -
3. Consider the Cauchy-Euler equation with k ∈ IR,
x2d2y
dx2+ x
dy
dx− k2 y = 0 .
(a) Show that x = 0 is a regular singular point.
(b) Use Frobenius’ method at at x0 = 0 to confirm the general solution
y = A1 xk + A2 x
−k .
(c) Under what conditions are the two solutions above linearly independent?
Continued . . .
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KMA354 Partial Differential Equations 3, 2010 - 5 -
4. The plot below gives contours of the real and imaginary components of
φ(z) = z3 = (reiφ)3 .
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
(a) Show this function to be analytic.
(b) Show that both the real and imaginary components are solutions to Laplace’s
equation.
(c) A potential function has the value zero long the lines y = 0 and y = +√
3x.
Along the line y(3x2−y2) = 10 the potential has value 60. Modify the com-
plex potential above and obtain a solution to Laplace’s equation between the
3 lines.
Continued . . .
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KMA354 Partial Differential Equations 3, 2010 - 6 -
5. Consider a string fixed at its endpoints, x = −π/2 and x = π/2 with initial
displacement U(x, 0) = cos(x) and zero initial velocity. All waves propagate
with unit wave speed.
(a) Solve the problem using separation of variables. Begin by rescaling the x
axis to aid the solution process.
(b) On an xt diagram indicate the ‘domain of dependence’ and ‘range of influ-
ence’ for the point (0.25, π).
END OF EXAM PAPER
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