8/13/2019 Engineering Mathematics Topic 4
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Tutorial 4 - Partial Differential Equation
1. Solve the following equations
a). )2t(x24xu 22
2
=
, given that at 0x = , t2eu = and t4
xu =
b). x2cose4yxu y2 =
, given that at 0y = , xcos
xu =
and at =x , 2yu = .
2. ftu
c1
xu
22
2
=
and 3c = , deter!ine the solution )t,x(f u = sub"e#t to
boundar$
#onditions, 0)t,0(u = and 0)t,2(u = for 0t
)x2(x)0,x(u = and 0tu
0t
=
= 2x0
%. & 'erfe#tl$ elasti# string is stret#hed between two
'oints 1( #! a'art. ts #entre 'ointis dis'la#ed 2 #! fro! its
'osition of rest at right angles to the original dire#tion of
the
string and then released with ero velo#it$. &''l$ing the
equation 22
22
2
t
u
c
1
x
u
=
with
1c 2 = , deter!ine the subsequent !otion )t,x(u .
4. *ne end & of an insulated !etal bar &+ of length 2 !
is e't at 0 while the otherend + is !aintained at 50 until a stead$
state of te!'erature along the bar isa#hieved. &t 0t = , the
end + is suddenl$ redu#ed to 0 and e't at that te!'erature.
sing heat #ondu#tion equationtu
c1
xu
22
2
=
, deter!ine an e/'ression for the
te!'erature at an$ 'oint in the bar distan#e / fro! & at an$
ti!e t.
0. The #entre 'oint of a 'erfe#tl$ elasti# string stret#hed
between two 'oints & and +, 4 !a'art, is defle#ted a distan#e
(.(1 ! fro! its 'osition of rest 'er'endi#ular to &+ and
released initiall$ with ero velo#it$. &''l$ the wave
equationtu
c1
xu
22
2
=
where
10c = to deter!ine the subsequent !otion of a 'oint P distant /
fro! & at ti!e t.
. &n insulated unifor! !etal bar, 1( units long, has
te!'erature of its ends !aintainedat 0 and at 0t = the te!'erature
distribution )x(f along the bar is defined b$
).x10(x)x(f = Solve the heat #ondu#tion equationtu
c1
xu
22
2
=
with 4c 2 = to
deter!ine the te!'erature u of an$ 'oint in the bar at ti!e
t.
. The ends of an insulated rod &+, 1( units long, are
!aintained at 0 . &t 0t = , thete!'erature within the rod rises
unifor!l$ fro! ea#h end rea#hing 2 at the !id-'ointof &+.
Deter!ine an e/'ression for the te!'erature )t,x(u at an$ 'oint in
the rod,distant / fro! the left-hand end at an$ subsequent ti!e
t.
Answer
+T+234(% Engineering athe!ati#s 5 Page 1 of 2
