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PDE801S - PARTIAL DIFFERENTIAL EQUATIONS - 2ND OPP - JULY 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 CODE: PDE 8015S
COURSE NAME: PARTIAL DIFFERENTIAL
EQUATIONS
SESSION: JULY 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 86
SUPPLEMENTARY/SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
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 8018
PARTIAL DIFFERENTIAL EQUATIONS
JULY 2022
QUESTION 1 [20 marks]
1. (a) From the following equations, form partial differential equations by eliminating the
arbitrary contacts g,h and 7.
i.
2e2y
z= gre!+
+h
ii.
z=g(at+y)+h(a—gyh)t ++7
(b) By eliminating arbitrary functions from the followings, form the partial differential
equation
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z= («—y)f(0? +9?)
ii.
f(x? —y?, vyz) = 0.
QUESTION 2 [25 marks]
2. (a) Solve the following differential equations by using Lagrange’s method
i.
(mz — ny)p + (nx — lz)q = ly — ma
ii.
(2? — y? — yz)p + (x? — y* — 2z)q = 2(x — y)
(b) Solve the following differential equations using Charpit method
i.
(+a )y = 92
oT
p=(z+qy)
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PDE 8015
PARTIAL DIFFERENTIAL EQUATIONS
QUESTION 3 [21 marks]
3. (a) Classify, reduce to normal form and hence solve
SUze + 10Ugy + 3Uyy = 0
(b) Classify, reduce to normal form and hence solve
Ure + 2Ugy + Uyy = 0
(c) Classify and reduce to normal form
Yo 2 Ure + LUyy = 0
JULY 2022
QUESTION 4 [20 marks]
4, (a) The temperature at one end of a 50cm long bar with insulated sides, is kept at 0°C
and that the other end is kept at 0°C' until steady-state condition prevails. The two
ends are then suddenly insulated and kept so. Find the temperature distribution
(b) Find the solution of the Cauchy problem
Ut — C’Ure = 0,
cER, t>0, u(z,0)=f(r), uw(z,0)=g9(z), cER.
(10)
End of Exam!