Una transformada nita de Hankel generalizada

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Z b
tf (t)Jν (tλ)dt
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Hν [f (t); λ] =
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x2
du
d2 u
+ (1 − 2l)x
+ [λ2 c2 x2c + (l2 − c2 ν 2 )]u = 0
2
dx
dx
+14
ν ≥ 0, l, c, λ > 0, x ∈ [0, a].
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&$@&:(& 6.6. @#'
u(x) = xl Jν (λxc ).
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hu(a) + u′ (a) = 0
+34
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T [u(x); ν, a; λi , l, c] = uν (λi , l, c) =
Z
0
a
x2c−l−1 u(x)Jν (λi xc )dx,
+!4
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!
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(h +
l − cν
)Jν (λac ) + λcac−1 Jν−1 (λac ) = 0.
a
"!#
8% 1-%"&'!-.% % *"6#-&% #&19 % % ,!f (x) =
∞
X
uν (λi , l, c)
i=1
!" #
!
kJν (λi )k =
(kJν (λi )k
xl Jν (λi xc )
"$#
a2c
l
ν2
[(h + )2 + (1 − 2 2c )]Jν2 (λi ac )
2c
a
λi a
"#$%&'( )*+),-.'.-(
% T [αf (x) + βg(x); ν, a; λi , l, c] = αf ν (λi , l, c) + βg ν (λi , l, c)
&% T [xcν+l ; ν, a; λi , l, c] =
'% T [f (px); ν, a; λi , l, c] =
acν
(cν
c2 λ2i
+ ha + l)Jν (λi ac )
1
T [f (x); ν, pa; pλci , l, c]
p2c−2l
(% T [Du(x); ν, a; λi , l, c] = a1−l Jν (λi ac )[hu(a) + u′ (a)] − λi 21 c 12 uν (λi , l, c)
)*+), Du ,- ,. */,01)*0
D(u) = x2−2c
du
d2 u
+ (1 − 2l)x1−2c
+ [(l2 − c2 ν 2 )]x−2c u
2
dx
dx
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d2 u
du
+ (1 − 2l)x1−2c ; ν, a; λi , l, c] =
2
dx
dx
Z a
2u
d
du
x2c−l−1 [x2−2c 2 + (1 − 2l)x1−2c ]Jν (λi xc )dx.
dx
dx
0
T [x2−2c
?,/101+)* 7+9,;01.,- -, 97,+,
Z a
Z a
du
d2 u
x−2l (xl Jν (λi xc ))dx.
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I=
dx
dx
0
0
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Z a
d
I = x1−l Jν (λi xc )u′ (x) −
u′ (x)x1−2l (xl Jν (λi xc ))dx.
dx
0
53'6*.'%&' $%&'()*%0, +,) +*)&'- -' 0')$6* #* -$(3$'%&' '7+)'-$8%
d
I = [x1−l Jν (λi xc )u′ (x) − x1−2l u(x) (xl Jν (λi xc ))]a0 +
dx
Z a
d
d2 u
u(x)x−1−2l [x2 2 (xl Jν (λi xc )) + (1 − 2l)x (xl Jν (λi xc ))]dx.
dx
dx
0
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Z a
l2 − c2 ν 2
I=M−
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]Jν (λi xc )dx,
x2c
0
0' 0,%0'
T [x2−2c
0,%0'
!"
I = M − λ2 c2 u − T [
(l2 − c2 ν 2 )
u(x)],
x2c
4 @%*#.'%&'
du
d2 u
+ (1 − 2l)x1−2c
+ (l2 − c2 ν 2 )x−2c u(x); ν, a; λi , l, c] = M − λ2 c2 u
2
dx
dx
M = a1−l Jν (λi ac )[hu(a) + u′ (a)].
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1 ∂2u
r2c−2 ∂ 2 u
(l2 − c2 ν 2 )
∂ 2 u (1 − 2l) ∂u
+
+ 2 2 =
−[
]u,
2
2
2
∂r
r
∂r
r ∂θ
α ∂t
r2
ν ≥ 0, l, c > 0. ;P=
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*/+$-2'/.+ #$
r2c−2 ∂ 2 u
(l2 − c2 ν 2 )
∂ 2 u (1 − 2l) ∂u
+
=
−
[
]u
∂r2
r
∂r
α2 ∂t2
r2
9<;
T
D
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α2 =
u(a, t) = h(t), t ≥ 0
u(r, 0) = f (r),
ut (r, 0) = g(r), 0 ≤ r ≤ a.
B#'/@+$%C+$&' 9<; -# *%#$#
2
1 ∂2u
∂u
2−2c ∂ u
=
r
+ (1 − 2l)r1−2c
+ (l2 − c2 ν 2 )r−2c u.
2
2
2
α ∂t
∂r
∂r
9 D;
E5,%(+$&' #, '5#/+&'/ */+$-2'/.+&+ F$%*+ &# G+$H#, @#$#/+,%C+&+ #$ 9 D; -# '=I
*%#$#
∂2u
= α2 a1−l Jν (λi ac )u(a, t) − a2 λ2i c2 u(λi , t).
∂t2
J-+$&' ,+ ('$&%(%8$ &# 2/'$*#/+3 -# &#&)(# ,+ -%@)%#$*# #()+(%8$ &%2#/#$(%+,
'/&%$+/%+ $' K'.'@#$#+
d2 u
+ a2 λ2i c2 u = a2 a1−l Jν (λi ac )h(t)
dt2
9
;
&'$&# λi (i = 1, 2, 3, ...) -'$ /+0(#- 5'-%*%L+- &# ,+ #()+(%8$ Jν (λi ac ) = 06
E, +5,%(+/ */+$-2'/.+&+ + ,+- ('$&%(%'$#- %$%(%+,#- -# *%#$#
u(λ, 0) = f (λ)
ut (λ, 0) = g(λ)
B#-',L%#$&' ,+ #()+(%8$ &%2#/#$(%+, K'.'@#$#+ +-'(%+&+ + 9
(%'$#- %$%(%+,#- -# '=*%#$#
; 1 )-+$&' ,+- ('$&%I
uc (λi , t) = f (λ) cos(caλi t) + (caλi )−1 g(λ) sen(caλi t).
9 M;
?+/+ (+,(),+/ ,+ -',)(%8$ ('.5,#.#$*+/%+ up -# K+(# )-'
N/'$-H%+$' N &#
R &#,
i
)dt
6 E 5+/*%/ &# +4)%
)9*; 1 &# ,+ /#,+(%8$ up (t) = y1 u1 + y2 u2 3 &'$&# yi = ( W
W
-# '=*%#$#
Z
αa1−l Jν (λi ac ) t
h(t) sen(caλi )(t − τ )dt.
9 O;
up (λi , t) =
cλ
0
!
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5 1 62 '(
αa1−l Jν (λi ac )
u(λi , t) = f (λ) cos(caλi t) + (caλi )−1 g(λ) sen(caλi t) +
cλ
Z t
h(τ ) sen(caλi )(t − τ )dt.
0
7,#*)8'#$'9 *3),&*#0% 1:2 (' $,'#'
∞
X
uν (λi , l, c)
xl Jν (λi xc ).
kJν (λi )k
i=1
;3),&*#0% )* &%#0,&,-# 0' </%#$'/* 5 3/%3,'0*0'( 0' )* <+#&,-# ='((') (' $,'#'
u(r, t) =
kJν (λi )k =
a2c 2
J (λi ac )
2c ν+1
>*(%( 3*/$,&+)*/'(? @*/* A($' 3/,8'/ /'(+)$*0% (' 8+'($/*# )%( (,.+,'#$'( &*(%(
3*/$,&+)*/'(?
B >%#(,0'/*#0% )*( &%#0,&,%#'( 0' </%#$'/* ' ,#,&,*)'(?
u(a, t) = 0, t ≥ 0,
u(r, t) = 2
u(r, 0) = f (r),
∞
X
i=1
R1
0
ut (r, 0) = 0,
0≤r≤a
rf (r)j0 (λi r)dr
cos(cαλi t)J0 (λi r).
J12 (λi ac )
!"#$%&' $"( %$) *+" ,-!."+/) *+ ") /+/.-)') +"( 0%$) $%-$#")- ,-+ +'0)*!
+' 12!3$+ 3 4%5-%/)6 78986 , :97;:9<=> 5)-) +" %?#%+'0+ $) ! + %'0-!*#$+'
@)"!-+ ,)-0%$#")-+ ) "! ,)-(/+0-! A#+ ?+'+-)"%B)' ") +$#)$%&' *+ !'*)>
C> D!' %*+-)'*!
0!/)'*!
ν E7>F6 "E7>F6 )E76 $E7>CF6+'>>>
f (r) = 26 g(r) = 16 h(t) = 2>
+ !.0%+'+
λi E<><G86
)*+/(
H' ,+-I" *+" /!@%/%+'0! *+ ") /+/.-)') + $!/! + /#+ 0-) +' ") I?#-) 7
J+ $!' %*+-) '#+@)/+'0+ ") +$#)$%&' 17K=
2
1 ∂2u
∂u
2−2c ∂ u
=
r
+ (1 − 2l)r1−2c
+ (l2 − c2 ν 2 )r−2c u.
2
2
2
α ∂t
∂r
∂r
5+-! + 0) @+B $!' ")
%?#%+'0+ $!'*%$%!'+ %'%$%)"+ 3 *+ L-!'0+-)M
hu(a, t) + u′ (a, t) = K(t), t ≥ 0, u(r, 0) = f (r), ut (r, 0) = g(r), 0 ≤ r ≤ a
4+ ,#+ *+ ),"%$)- ") 0-)' L!-/)*) ?+'+-)"%B)*) *+ N)'O+" 3 0+'+- +' $#+'0) ")
$!'*%$%&' *+ L-!'0+-) + 0%+'+
d2 u
+ a2 λ2i c2 u = a2 a1−l Jν (λi ac )K(t)
dt2
!
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a
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u(λi , t) = f (λ) cos(caλi t) + (caλi )−1 g(λ) sen(caλi t) +
×
cλ
Z t
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0
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a
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