Page 382 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 14] FOURIER INTEGRALS 373

10 @ x < 1
0 x A 1
14.18. If f ðxÞ¼ find the (a) Fourier sine transform, (b) Fourier cosine transform of f ðxÞ.In
each case obtain the graph of f ðxÞ and its transform.
r ffiffiffi r ffiffiffi
2 1 cos 2 sin
Ans: ðaÞ ; ðbÞ :

x
14.19. (a)Find the Fourier sine transform of e , x A 0
ð
1 x sin mx
ðbÞ Show that 2 dx ¼ e m ; m > 0byusing the result in ðaÞ:
0 x þ 1 2
(c) Explain from the viewpoint of Fourier’s integral theorem why the result in (b) does not hold for m ¼ 0.
p ffiffiffiffiffiffiffiffi 2
Ans. (a) 2= ½ =ð1 þ ފ
14.20. Solve for YðxÞ the integral equation
8
1 0 @ t < 1
ð
1 <
2 1 @ t < 2
YðxÞ sin xt dx ¼
0 : 0 t A 2
and verify the solution by direction substitution.
Ans. YðxÞ¼ð2 þ 2cos x 4cos 2xÞ= x
PARSEVAL’S IDENTITY
ð ð 2
1 dx 1 x dx
14.21. Evaluate (a) ; by use of Parseval’s identity.
2 2 ðbÞ 2 2
0 ðx þ 1Þ 0 ðx þ 1Þ
x
[Hint: Use the Fourier sine and cosine transforms of e , x > 0.]
Ans: ðaÞ =4; ðbÞ =4
2
ð ð 4
1 1 cos x 1 sin x
14.22. Use Problem 14.18 to show that (a) dx ¼ ; ðbÞ 2 dx ¼ .
0 x 2 0 x 2
ð 2
1
14.23. Show that ðx cos x sin xÞ dx ¼ .
6
0 x 15
MISCELLANEOUS PROBLEMS
2
@U @ U x
14.24. (a) Solve ¼ 2 , Uð0; tÞ¼ 0; Uðx; 0Þ¼ e ; x > 0; Uðx; tÞ is bounded where x > 0; t > 0.
@t @x 2
(b)Give a physical interpretation.
2
2 t
ð
2 1 e sin x
Ans: Uðx; tÞ¼ d
2
þ 1
0
2
@U @ U x 0 @ x @ 1
14.25. Solve ¼ ; U x ð0; tÞ¼ 0; Uðx; 0Þ¼ , Uðx; tÞ is bounded where x > 0; t > 0.
@t @x 2 0 x > 1
ð
2 1 sin cos 1 t
2
Ans: Uðx; tÞ¼ þ 2 e cos xd
0
14.26. (a) Show that the solution to Problem 14.13 can be written
p
p
2 ð x=2 t ffiffi v 2 1 ð ð1þxÞ=2 t ffiffi v 2
e e dv
Uðx; tÞ¼ p ffiffiffi dv p ffiffiffi
p ffiffi
0 ð1 xÞ=2 t