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PDE801S - PARTIAL DIFFERENTIAL EQUATIONS - 1ST OPP - JUNE 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of science Honours in Applied Mathematics
QUALIFICATION CODE: 35BAMS
LEVEL: 8
COURSE c CODE: PD ESO
COURSE NAME: PARTIAL DIFFERENTIAL
EQUATIONS
SESSION: JUNE 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 90
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Prof A. S. EEGUNJOBI
MODERATOR:
Prof O.D. MAKINDE
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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PDE 80158
PARTIAL DIFFERENTIAL EQUATIONS
JUNE 2022
QUESTION 1 [25 marks]
1. (a) From the following equations, form partial differential equations by eliminating the
arbitrary contacts g,h and j.
1.
g?eY
z= gue’ +
+h
a
il.
z=g(a+y)+h(a—y)+ght+)
(b) By eliminating arbitrary functions from the followings, form the partial differential
equation
i.
z= («—y)f(x* +4’)
ii.
f(z? — y?, zyz) = 0.
QUESTION 2 [20 marks]
2. Solve the following first order PDE
(a) (y? + 2°) — vy +2z=0
(b) (z+ 2)% + (yt+z)% —x—y=0
((Ac)) 2((a?y +— 2z)y5?2)8+ =y(zay—! 2+) 22= 2(t —y)
QUESTION 3 [25 marks]
3. (a) Classify, reduce to normal form and hence solve
3Uge + 10Ugy + 3uUyy = 0
(b) Classify, reduce to normal form and hence solve
Une + 2Ugy + Uyy = 0
(c) Classify and reduce to normal form
YF2 Ug+ 2 2 Uyy= 0
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PDE 8015S.
PARTIAL DIFFERENTIAL EQUATIONS
JUNE 2022
QUESTION 4 [20 marks]
4. (a) Solve
t1odau ddyu _~°
given that u(0,y) = 12e~°#.
(10)
(b) Find the solution of the Cauchy problem
Utt — Cte = 0,
reER, t>0, u(z, 0) = f(z), uz(z, 0) = g(x), rER.
(10)
End of Exam!