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[0, a]/ 2 λi (i = 1, 2, ...) &!" -%7)#& ,!&*1*6%& # $% #)(%)*4" (h + l − cν )Jν (λac ) + λcac−1 Jν−1 (λac ) = 0. a "!# 8% 1-%"&'!-.% % *"6#-&% #&19 % % ,!f (x) = ∞ X uν (λi , l, c) i=1 !" # ! kJν (λi )k = (kJν (λi )k xl Jν (λi xc ) "$# a2c l ν2 [(h + )2 + (1 − 2 2c )]Jν2 (λi ac ) 2c a λi a "#$%&'( )*+),-.'.-( % T [αf (x) + βg(x); ν, a; λi , l, c] = αf ν (λi , l, c) + βg ν (λi , l, c) &% T [xcν+l ; ν, a; λi , l, c] = '% T [f (px); ν, a; λi , l, c] = acν (cν c2 λ2i + ha + l)Jν (λi ac ) 1 T [f (x); ν, pa; pλci , l, c] p2c−2l (% T [Du(x); ν, a; λi , l, c] = a1−l Jν (λi ac )[hu(a) + u′ (a)] − λi 21 c 12 uν (λi , l, c) )*+), Du ,- ,. */,01)*0 D(u) = x2−2c du d2 u + (1 − 2l)x1−2c + [(l2 − c2 ν 2 )]x−2c u 2 dx dx "2# " #3"&#3"'# -, ),)45,+ 6157.8,+9, 4-1+)* "(# : 1/.751+)* 1),541)18,+9, 1.;4+10,.157*+,- ), 0,5400,+571 ), .1- 64+57*+,- ), <,--,.% /*%-0' .- 12 =*+-7),0,8*- .*- )*- /078,0*- 9>087+*- )76,0,+571.,- ), "2# d2 u du + (1 − 2l)x1−2c ; ν, a; λi , l, c] = 2 dx dx Z a 2u d du x2c−l−1 [x2−2c 2 + (1 − 2l)x1−2c ]Jν (λi xc )dx. dx dx 0 T [x2−2c ?,/101+)* 7+9,;01.,- -, 97,+, Z a Z a du d2 u x−2l (xl Jν (λi xc ))dx. x1−2l 2 (xl Jν (λi xc ))dx + (1 − 2l) I= dx dx 0 0 ! !"#$%&''() "! &*+,-. / 0! 1-'+,2-. 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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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u(r, t) 0' 3%* .'.?)*%* D3' '- 0'F,).*0* -$./&)$:*.'%&'C J# 0'-+#*O A*.$'%&, u(r, t) '-&B .,0'#*0, +,) #* ':3*:$8% 0' ,%0* %, H,.,(/%'* 1 ∂2u r2c−2 ∂ 2 u (l2 − c2 ν 2 ) ∂ 2 u (1 − 2l) ∂u + + 2 2 = −[ ]u, 2 2 2 ∂r r ∂r r ∂θ α ∂t r2 ν ≥ 0, l, c > 0. ;P= Q3+,%(*.,- D3' #* .'.?)*%* &$'%' )*0$, *2 '-&B )E($0*.'%&' @I* *#)'0'0,) 0' -3 :$):3%F')'%:$* 4 D3' $%$:$*#.'%&' ,:3+* 3%* +,-$:$8% 0'-+#*A*0* $%0'+'%0$'%&' 0' #* 6*)$*?#' *%(3#*) θ2 * +*)&$) 0' #* :3*# -' #' $.+)$.' 3%* 6'#,:$0*0 $%$:$*# '% &RSC !" #$"!%&'$(")" *!+#" ), -"!.,/ 0,!,$"/+1")" ! "#$%#$&' #$ ()#$*+ ,+ -%.#*/0+ (%/(),+/ &# ,+- ('$&%(%'$#- %$%(%+,#- 1 &# 2/'$*#/+3 -# &#&)(# 4)# ) #- %$&#5#$&%#$*# &# θ6 7$ #-*# (+-' ,+ #()+(%8$ &%2#/#$(%+, 9:; -# */+$-2'/.+ #$ r2c−2 ∂ 2 u (l2 − c2 ν 2 ) ∂ 2 u (1 − 2l) ∂u + = − [ ]u ∂r2 r ∂r α2 ∂t2 r2 9<; T D &'$&# T #- ,+ *#$-%8$ &# ,+ .#.=/+$+ 1 D #- ,+ .+-+ 5'/ )$%&+& &# >/#+ &# ,+ .#.=/+$+6 ?+/+ *#$#/ )$+ -',)(%8$ .>- @#$#/+, &#, 5/'=,#.+ ('$-%&#/#.'- ,+- -%@)%#$*#('$&%(%'$#- &# 2/'$*#/+ # %$%(%+,#-A α2 = u(a, t) = h(t), t ≥ 0 u(r, 0) = f (r), ut (r, 0) = g(r), 0 ≤ r ≤ a. B#'/@+$%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 ! !"#$%&''() "! &*+,-. / 0! 1-'+,2-. "#$%#&'( )* (%)+&,-# .'#'/*) 0' 1 2 $'#,'#0% '# &+'#$* )* (+3'/3%(,&,-# 0' 1 42 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 ! !"#$%&''() "! &*+,-. / 0! 1-'+,2-. "#$"% λi (i = 1, 2, 3, ...) &#$ '()*%& +#&,-,.(& "% /( %*0(*,1$ (h + l − cν )Jν (λi ac ) + λcac−1 Jν−1 (λac ) = 0. a 2&($"# 0$ +'#*%",3,%$-# &,3,/(' (/ ($-%',#' &% #4-,%$% αa1−l Jν (λi ac ) u(λi , t) = f (λ) cos(caλi t) + (caλi )−1 g(λ) sen(caλi t) + × cλ Z t K(τ ) sen(caλi )(t − τ )dt. 0 5+/,*($"# /( -'($&6#'3("( ,$.%'&( &% #4-,%$% 0$( &#/0*,1$ 4(&-($-% 7%$%'(/ "%/ +'#4/%3( "% /( 3%34'($(8 "("( +#' u(r, t) = ∞ X uν (λi , l, c) i=1 (kJν (λi )k xl Jν (λi xc ) 9%$,%$"# %$ *0%$-( /( *#$",*,1$ "% 6'#$-%'( : +'#+,%"("%& "% /( 60$*,1$ 4%&&%/ &% -,%$% l ν2 a2c [(h + )2 + (1 − 2 2c )]Jν2 (λi ac ) kJν (λi )k = 2c a λi a ;(&# +('-,*0/('< ;0($"# ν = 1,258 l = 0,88 a = 28 c = 1,258 &% #4-,%$% λi = 3,857 : ("%3=& &% *#$&,"%'($ f (r) = 08 g(r) = 18 K(t) = 0>?$ /( @70'( &% 30%&-'( 0$ +%'@/ "% .,4'(*,1$ "% /( 3%34'($(> !"!#!$%&'( ABC 5/DE(F', ($" G(//(8 H> I>< J$ ($ ,$-%7'(/ -'($&6#'3 ,$.#/.,$7 K%&&%/ 60$*-,#$&8 K0//> H#*> L(-M> L(*%"#$,%8 N O !!PQ8 RSBN> A C T> 5/, ($" G(//(8 H> I>< 5 7%$%'(/,U%" E($V%/ -'($&6#'3 ($" ,-& 0&% 6#' &#/.,$7 *%'-(,$ +('-,(/ ",W%'%$-,(/ %X0(-,#$&8 Y 50&-> L(-M> H#*> PBK OBZZZQ8 B!RSBB[> A\C 5$"'%]&8 ^> ?>8 5&V%:8 _> ($" _#:8 _>< H+%*,(/ `0$*-,#$&8 BZZZ8 ;(34',"7% 2$,.%'&,-: a'%&&8 b%] c#'V> APC K#:*%8 d> ?> ($" e,a',3(8 _> ;>< ?*0(*,#$%& e,6%'%$*,(/%& : +'#4/%3(& *#$ .(/#'%& %$ /( 6'#$-%'(8 BZ[Z8 -%'*> %">8 ?",-> I,30&(8 L%f,*#8 BZ[Z> 8 !![> ARC e%4$(-M8 I> ($" KM(--(8 e>< T$-%7'(/ -'($&6#'3& ($" -M%,' (++/,*(-,#$&8 !![ K0//> H#*> L(-M> L(*%"#$,%8 &%*#$" %">8 ;M(+3($ ($" E(//g;_; a'%&&8 K#*( _(-#$8 `I> BZ! 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