…2/- Final Examination
2017/2018 Academic Session May/June 2018
JIM417 – Partial Differential Equations [Persamaan Pembezaan Separa]
Duration : 3 hours [Masa: 3 jam]
__________________________________________________________________________________________
Please ensure that this examination paper contains EIGHT printed pages before you begin the examination.
Answer ALL questions. You may answer either in Bahasa Malaysia or in English.
Read the instructions carefully before answering.
Each question is worth 100 marks.
In the event of any discrepancies, the English version shall be used.
Sila pastikan bahawa kertas peperiksaan ini mengandungi LAPAN muka surat yang bercetak sebelum anda memulakan peperiksaan ini.
Jawab SEMUA soalan. Anda dibenarkan menjawab sama ada dalam Bahasa Malaysia atau Bahasa Inggeris.
Baca arahan dengan teliti sebelum anda menjawab soalan.
Setiap soalan diperuntukkan 100 markah
Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah digunapakai.
1. (a). The Heavisde function is defined by
c t
c t c
t U
, 1
, , 0 ) (
By using the definition of Laplace Transform, prove that L
U(tc)f(tc)
ecsF(s)where
L
f(t) F(s).(40 marks) (b). Using Laplace Transform, solve the initial value problem,
utt 2u0, 0, t0 subject to the conditions
) 0 (
, 0 ) 0 ( ut
u
(60 marks) 2. Consider a given function
0, 2
( ) 1,
2 2
0, 2
x
f x x
x
(a). State the precise numerical value of f (x) for each x in the interval
−π ≤ x ≤ π.
(20 marks) (b). Compute the Fourier coefficients aj , bn for f (x).
(50 marks) (c). Using the fact that
0x f t dt x( )
for 2 x 2, integrate the Fourier series for f (x) to obtain the expansionsin(2 1) , )
1 2 (
) 1 (
4 1 2
1
x k k
x
k
k
.
2 2
x (30 marks)
…4/- 3. Given a partial differential equation
u u u 0
x t
, x0, t0, with boundary and initial conditions
(0, ) 0 0,
u t t and u x( , 0) sin( ), x x0. (a). By using Laplace transform, show that
( 1) 2
( 1)sin( ) cos( ) e ( , )
2 2
s x
s x x
U x s
s s
.
(60 marks) (b). Find the inverse Laplace transform of (a).
(40 marks) 4. (a). Classify each of the following partial differential equations as
hyperbolic, elliptic, or parabolic:
(i). uxx2uxyuyyuxuy 0 (ii). uxx2uxy2uyyuxuy sin( )xy (iii). 2uxx4uxy6uyyux 0
(30 marks)
(b). Find the canonical form of the following hyperbolic partial differential equations. Be sure to show the change of coordinates that reduces the partial differential equations to canonical form
6 16 0
xx xy yy
u u u .
(70 marks) 5. Find the solution to the heat conduction problem
2 , 0 , 0 (0, ) 0
( , ) 0
( , 0) 3sin 5 ( ).
2
t xx
x
u u x t
u t
u t
u x x f x
(100 marks)
1. (a). Fungsi Heavisde function ditakrifkan oleh
c t
c t c
t U
, 1
, , 0 ) (
Dengan menggunakan takrifan Jelmaan Laplace, buktikan bahawa L
U(tc)f(tc)
ecsF(s)di mana
L
f(t) F(s).(40 markah) (b). Dengan menggunakan jelmaan Laplace, selesaikan masalah nilai
awal-sempadan
0 , 0 ,
2 0
u t
utt Tertakluk kepada syarat
(0) 0,
(0) .
t
u
u
(60 markah) 2. Pertimbangkan fungsi yang diberikan
0, 2
( ) 1,
2 2
0, 2
x
f x x
x
(a). Nyatakannilaiberangkatepatf (x)bagisetiap x dalamselang −π ≤ x ≤ π.
(20 markah) (b). Kirakan pekali Fourier aj , bn kepada f (x).
(50 markah) (c). Menggunakan fakta
0x f t dt x( )
kepada 2 x 2 , kamirkan siri Fourier untuk f (x) untuk dapakan pengembangan.sin(2 1) , )
1 2 (
) 1 (
4 1 2
1
x k k
x
k
k
.
2 2
x (30 markah)
…6/- 3. Diberi persamaan pembezaan separa
u u u 0
x t
, x0, t0, dengan syarat sempadan dan syarat awal
(0, ) 0 0,
u t t dan u x( , 0) sin( ), x x0.
(a). Dengan menggunakan transformasi Laplace, tunjukkan bahawa
( 1) 2
( 1)sin( ) cos( ) e ( , )
2 2
s x
s x x
U x s
s s
.
(60 markah) (b). Cari transformasi Laplace songsang bagi (a).
(40 markah) 4. (a). Klasifikasi setiap persamaan pembezaan separa berikut sebagai
hiperbolik, eliptik, atau parabola:
(i). uxx2uxyuyyuxuy 0 (ii). uxx2uxy2uyyuxuy sin( )xy (iii). 2uxx4uxy6uyyux 0
(30 markah) (b). Cari bentuk berkanun persamaan pembezaan separa hiperbola yang berikut. Pastikan untuk menunjukkan perubahan koordinat yang mengurangkan persamaan pembezaan separa kepada bentuk berkanun
6 16 0
xx xy yy
u u u .
(70 markah) 5. Cari penyelesaian kepada masalah konduksi haba
2 , 0 , 0 (0, ) 0
( , ) 0
( , 0) 3sin 5 ( ).
2
t xx
x
u u x t
u t
u t
u x x f x
(100 markah)
Formulae
x = x x
u u u
y = y y
u u u
2 2
= 2
xx x x x x xx xx
u u u u u u
xy = x y x y y x x y xy xy
u u u u u u
2 2
= 2
yy y y y y yy yy
u u u u u u .
o=1
= + cos + sin
2 n n n
a nπx nπx
f x a b
L L
with
L
o L
a 1 f x dx L
1
cos , 1, 2, 3, ...
L n
L
a f x n x dx n
L L
L
1 sin , 1, 2, 3, ...
L n
b f x n x dx n
L L
o1
2 n n cos
a n x
f x a
L
with
0
2 cos , 1, 2, 3, ...
L n
a f x n x dx n
L L
1 nsin
n
f x b n x
L
…8/- with
0
2 , 1, 2, 3, ...
L n
b f x n x dx n
L L
12
inx
f x c en
with
1
, 0, 1, 2, ...
inx
cn f x e dx n
2 2
2 0 has solution
d y y
dx
or cosh sinh
x x
y Ae Be C x+ D x.
2 2
2 0 has solution
d y y
dx
y = A cos x + B sin x.
2 2 2
2 0
d R dR
r r n R
dr dr has solution
n n
n n n
R C r D
r
2
2 0 has solution
d R dR
r r
dr dr R = A + Bn r.
L
e f tt
F s
.L
H t a
esasL f t a H t a e F sas
L
fn
t s F sn
sn1f
0 sn2f
0 ... sfn2
0 fn1
0L n 1 n nn
t f t d F s
ds
L
0
t F s
f u du s
L
1
0 t
F s G s f u g t u du f g
Laplace Transforms
f(t)
L f t F s
1
tn, n = 1, 2, 3, …
eat
cos at
sin at cosh at
sinh at
t cos bt
t sin bt
eat cos bt
eat sin bt
1 s
1
!
n
n s
1 s a
2 2
s s a
2 2
a s a
2 2
s s a
2 2
a s a
2 2
2 2 2
s a s b
s22bsb2
2
2 2s a
s a b
2 2b s a b - oooOooo -