v ( x;t ) f ( x;t ) x 2 › t 2 IR v (x,t) v (x,t) xx0L W › ‰ IR ›=]0 ;L [ L

(1)

9:<;=>=?&A@BC/)')DA&-/)?*&DAE&:%FD%G;=D

HIKJMLBN$OQPMRSOTRSUMVWO=X P$YZP\[^]`_aX bQ]dcBUfeg]hXibQ]BLdUkj&]BLdPMl[^]dUdmnLdb!OT]imnogPqp

rs$tu<vdwskxasqyxCz{1s |}{ xu<v ~Sz1kx|vh{Wgkxz1~s Wdx1kWdx| z1s.skx{ zku|Kz1s.~sqyCt1k{1vhk{1sqyai|

udx<vh|}u<sqyt1k{1vhk{1sqy z|vh{ xz{1s|}&t!vhuxd{1qsCqvh{1y|~ku sC~1d{1ysqyy<k|sk{1qsqy>~s|}{k{|skzu

du|}y|}{ x<skui|sk{{1sk{ x~1d{1y?~s,xu<qy?{1vh u<skz1y<sqyCy|}xzdx|vh{1y?tudx| z1sqy$u<qy<|yxd{1qs$dzsk{ xC~sqy

vhz udsqyf~=duxhqvhz1y<x|Kz1sy<x

|}}|}x<~sqy`Kk|kzsqyf~sxud{1yt!vhu<xu<sq1sku<1s~s|}ysksk{ x<y

t!kxu<vh}|sku<yT}}St!vhzu{Tsk{k|}x<skuCKz1s, z1sk z1sqygz{1sqy

?¡Q¢T£T¤¥¡=¦¨§1©«ª=¬^«¢T¤

®C{qvh{1y|~ku<s,sqy?~skzS¯°y±Syx<ksqyZy<z|}hd{ x<y$

²

z{1sqvhu<~sk³hyx|Kz1sx<sk{1~Szs~svh{z1skzu

L

~vh{ x³´qvh{Sµ1zudx|vh{¶~su<\·ku<sk{1qssqy<x

~qku|}x<s¸tduZs¸y<sksk{ x

Ω =]0, L[

^¹

²

z{1ssk ud{1s¸k³hyx| z1sx<sk{1~Szsh1~vh{ x³.qvh{Sµ1zudx|vh{~su<\·ku<sk{1qs,sqyxZ~qku|}x<satdus

~vhWd|}{1s

vhu{1

Ω ⊂ IR ²

v (x,t)

º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º°º

»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»°»

Ω

x x 0

L

®C{¼{1vhx<s

x ∈ Ω

³hdu<| }s½~=sqythqs°skx

t ∈ IR ⁺

³hdu|³ s½~sx<sk&t1y)skxvh{¼v ~k}|y<sfs

~kt³hqsksk{ xCxud{1y<sku<y<s

v(x, t)

~s¸qsqy?y<±iyx<ksqy¾sk{½u<kt!vh{1ys .z{1s¸s\¯k|}xdx|vh{

f (x, t)

^tdu?s

(2)

(I)



 

 

 

 

∂ ² v

∂t ² (x, t) − ∆v(x, t) = f(x, t)

^yzu

Ω × IR ^+∗

v(x, t) = 0

^y<zu

∂Ω × IR ^+∗

v(x, 0) = 0

^y<zu

Ω

∂v

∂t (x, 0) = 0

^y<zu

Ω.

®C{qvh{1y|~ku<s,s¸hy~=z{1s¸s\¯k|}xdx|vh{duvh{|Kz1s$~s,³8·Àvhus

f (x, t) = g(x) cos(ωt)

vhÁ

ω ∈ IR

sqyxZz{1s,tzydx|vh{skx

g

z{1sa·vh{1kx|vh{~vh{{1qs$~s

L ² (Ω)

®C{Â{1vhx<s

(λ _k , u _k ) _k∈IN ^∗

sqykksk{ x<ytu<vhtu<sqy~SzÂ³dt³hk|sk{Ã~sÄ|}u|skxfyzuWs~vhWd|}{1s

Ω

vhu<~vh{{1qy,~s x<sk}s8yvhux<s8Kz1s

λ 1 < λ 2 ≤ λ 3 ≤ · · ·

®C{udtt!sk}s½ÅÀ\·«Æ

¿

!Ç1qvhu<ksWÈiÉSÊBËÍÌCKz1s

³$y<z|}x<s

(λ _k ) _k∈IN ^∗

x<sk{1~`Ksku<y

+ ∞

z1s?sqy

(u _k ) _k∈IN ^∗

·Àvhusk{ xz{1s hys?|} skux|sk{{1s,~s

L ² (Ω)

skx z1sat!vhzux<vhzx

k ∈ IN ^∗

( u _k ∈ H ₀ ¹ (Ω),

− ∆u _k = λ _k u _k .

®C{¼udtt!sk}s°sk{Sµ1{ÎKz1sf~1d{1y&s°hy&~sf³qvhu<~sÏÅ

Ω =]0, L[

^Ì\«sqy

(λ _k , u _k ) _k∈IN ^∗

y<vh{ xqvh{{ z1y

d{d}± x| z1sksk{ x

λ _k = k ² π ²

L ² , u _k (x) = r 2

L sin µ

kπ x L

¶ .

®C{´{1vhx<s

(g _k ) _k∈IN ^∗

sqy8qv s\ÐWk|sk{ x<y.~sW³f·vh{1kx|vh{

g ∈ L ² (Ω)

~1d{1y ³ hy<s

(u _k ) _k∈IN ^∗

^{skx vh{}

t!vy<s

ω _k = √

λ _k

^¨Ñz{|Kzsy<vh}zx|vh{Ï~s

(I)

^~1d{1y

C ⁰ ([0, + ∞ [, H ¹ (Ω)) ∩ C ¹ ([0, + ∞ [, L ² (Ω))

ÅÀ\·¨Æ

¿

Ç1qvhu<ks8ÒSÉSÓBË^Ìt!skzx?yqku|}u<s¸yvhz1yZ³ ·vhus

v(x, t) =

+∞

X

k=1

y k (t) u k (x).

Ô&ÕÖK×Ø¾Ù^Ú!ÛGÜÝÞ x }|}u`z{1sqKzdx|vh{ß~S|à=ku<sk{ x|sk}sK\u<|á·qs½tdu`³·Àvh{1kx|vh{

y _k (t)

^ÅÀ~s

IR ⁺

hdskzu<y?~1d{1y

IR

Ì\Stz|yu<qy<vhz1~Su<s d{d}± x| z1sksk{ xCqskxx<s¸q zdx|vh{¨

Ô&ÕÖK×Ø¾Ù^Ú!Û¶âQÝDã

zTh~ |sk{ xg|}Dvhu<y< z1s$³&·Íu<q z1sk{1qs8~=s\¯Sk|}xdx|vh{

ν = _2π ^ω

sqyxCKvh|}y<|}{1s ~=z{1s8~sqy

·Íu<q z1sk{1qsqy&tu<vhtu<sqy

ν _k = ^ω _2π ^k

~Sz¼y<±iyx\sä°Äaqku|}u<s°skx&{1vh&skusqy.t1k{1vhk{1sqyWt ±iy| z1sqy

qvhuu<sqyt!vh{1~1d{ x<y

Ô&ÕÖK×Ø¾Ù^Ú!ÛÂå=Ý=æ vhzu

K ∈ IN ^∗

vh{½{1vhx<s

v K (x, t) =

K

X

k=1

y _k (t) u _k (x).

ç vh{ xus\uCKz1s

k v( · , t) − v _K ( · , t) k L ² (Ω) ≤ 2





+∞

X

k=K+1

| g _k | ² (ω _k ² − ω ² ) ²





1/2

.

(3)

sup

t∈IR ⁺ k v( · , t) − v _K ( · , t) k L ² (Ω) −→

K→+∞ 0.

®C{&Wd|}{ x<sk{d{ xkx

}|}u>Kzst}z1y³C·vh{1kx|vh{

g

sqyx)u<kz}|ku<st}z1y)³i|}x<sqy<y<sZ~sZqvh{ Kskusk{1qs

~s

(u _K ) _K∈IN ^∗

^Ksku<y

u

sqyx?kskqs

Ô&ÕÖK×Ø¾Ù^Ú!Û¶éTÝ ®C{¥yztt!vy<s&~1d{1y,qskxx<s. z1sqyx|vh{¥ z1s8s.yztt!vhux¸~s

g

sqy<xz{¥qvh&thkx¸|}{1k}z1y

~1d{1y

Ω

^skx8Kz1s

g ∈ H ^m (Ω)

t!vhzu z{êsk{ x|sku

m ≥ 0

ÅsqW³qvh{ sk{ix|vh{

H ⁰ (Ω) = L ² (Ω)

^Ì\

ç vh{ xus\uCKz1s

+∞

X

k=1

λ ^m _k | g k | ² < + ∞ .

Þ {~q~Sz|}u<s$Kz1s¸y|

g ∈ H ^m (Ω)

sqyxC.yztt!vhux?qvh&thkx

sup

t∈IR ⁺

k v( · , t) − v _K ( · , t) k _L ² _(Ω) ≤ C λ ^m/2+1 _K+1 ,

vhÁ

C

~kt!sk{1~~s

g

ω

^skx

m

Wd|}y{1s¸~kt!sk{1~°t*hy~s

K

æ vhzux<sku&|}{1skuaqskxx<s8kxz1~s8~s qvh{ Ks\u<s\{ks!|}u<sqyx<s8Wsqyx|}sku³i|}x<sqy<y<s$~s ku<vh|y<yMd{1qs ~s$³

yz|}x<s

(λ _k ) _k∈IN ^∗

®C{½t!skzxvh{ xu<skuCKz1sat!vhzu?z{½vhzKskux

Ω ⊂ IR ^d λ K _K→+∞ ∼ 4 π ²

µ K

mes(B _d ) mes(Ω)

¶ 2/d

Å ¿ Ì

vhÁ

B _d = ⁿ x ∈ IR ^d , | x | < 1 ^o

Ô&ÕÖK×Ø¾Ù^Ú!Û-ë=Ýì ku|µsku`Kz1s°³·Àvhu zs´Å

¿

Ì ·vh{1kx|vh{{1s½t!vhzu

Ω =]0, L[

)Äakvh{ xu<sku`sk{1y<z|}x<s

qskxx<sa·vhu zs,t!vhzug ± t!sku<kz s

Ω =]0, 1[ ^d

^~s

IR ^d

íMî*ïhðñ<òóÀðÀôhîõ«öî÷<ñkø\ð^øò.ùiî!ú,ú<ûMü*ø<úMýMýMðôhîòhî*òhþáÿóÀðqùú,ïKúMýBòhþú\ùiøMýü*ø<ôMü*ø<úMý,úkóôhîñ<ôWú\î!ñúkøòCüòhø

ôhîóÀø<ú\økù1úaþúî!ô \ø<ú

N (µ)

ïKúBòhþú\ùiøMý>ü!øôMü*ø<úMýðÍî÷\ø\ðú\ùiø<úMý,ôhù½÷òhþúMý

µ

^d÷\ø\ðúaþòqôhø8ùiþú ïKú.úkÿdþ

N (µ) _µ→+∞ ∼ mes(B _d )

(2π) ^d mes(Ω) µ ^d/2 .

$ÚÖ ÛQØÙdÖ1ÝSÞ {uhy<y<sk ³d{ xsqyDu<qyz}xdx<y)tukqq~sk{ xyvh{v x|sk{ x z1sy|

Ω

sqyx)z{&vhzKskux

~s

IR ^d

^skx?y|

g ∈ H ^m (Ω)

sqy<x?.yzt1t!vhux?qvh&thkx*dvhu<yZt!vhzu

K

hy<y<s ¸ud{1~=

sup

t∈IR ⁺ k v( · , t) − v _K ( · , t) k L ² (Ω) ≤ C K ^(m+2)/d

vhÁ

C

{1s&~kt!sk{1~thy$~s

K

^Þ {tdux|kz}|skuy|

g ∈ D (Ω)

=³kxvK~1s~1s.u<qy<vh}zx|vh{´~s

(I)

tduyzt!skut!vy<|x<|}vh{~sav ~sqyqvh{ Kskus,t}z1yZ |}x<s, z1sa{¨|}&t!vhux<s¸ z1sk}satz|y<yMd{1qsa{1kKdx|}Ks,~s

K

^>®C{ ~S|}xWKz1s°qskxx<s°kx1v ~ssqyx~=vhu<~Su<s°|}{Sµ1{|g"!vhx<vh{1yqskt!sk{1~1d{ xW z1s°qskxx<s° |}x<sqyy<sf~s

qvh{ skusk{1qsf{1s`t!skzx$#kxu<sfv x<sk{ z1sfsk{êtudx| z1s z1s`y<|gvh{ qvh{{&%x&s\¯1hkx<sksk{ x.sqy8v ~sqy

tu<vhtu<sqya~Sz³dt³hk|sk{¥~s8Ä|}u|skx,yzu

Ω

!qs z|{¨sqyxs8hyaKz1s t!vhzua~sqy~vhWd|}{1sqy

Ω

^xu<qy

tdux|kz}|sku<y8ÅÀ~S|}sk{1y|vh{ ¿ t¸~s

IR ^d

~S|yKz1s,~s

IR ²

=}}nÌ\

(4)

Ô&ÕÖK×Ø¾Ù^Ú!Û:9=ÝÞ ku|}u<sz{´tu<vhud&s<;>=@?BAC0Dt!skuskxxd{ x&~s`y|} zsku8³u<kt!vh{1y<s~=z{1sqvhu<~s

~s&vh{z1skzu

L = 1

^Å

Ω =]0, 1[

Ì,fgs\¯Sk|}xdx|vh{

f (x, t) = χ _]a,b[ (x) cos(ω t)

^ÅÀvhÁ

χ _]a,b[

~qy|}{1s

³·vh{1kx|vh{¥duhkx<ku|yx| z1s&~s g|}{ x<skud}s

]a, b[

Ì\Qyzuz{|}{ x<skuhd}s&~s x<sk&t1y

[0, T ]

=rs8tu<vd

ud&s~skiut!skuskxxu<s~si|yzd}|y<skuyzu`z{ßE#ksudt|Kz1sgkKvh}zx|vh{ dzÃqvhzu<y~Sz

x<sk&t1y?~saxu<vh|y?y<vh}zx|vh{1yCdttu<v 1qsqy

¿

³&y<vh}zx|vh{{ zku| z1s¸v

xs\{iz1s¸sk{dtt}| zd{ x?³.kx1v ~s$~s¸yzt!skut!vy|}x|vh{~s,v ~sqy

ÅÀ~1kqvh&t!vy<|}x|vh{y<zuZsqy

K

tu<sk&|sku<yv ~sqytu<vhtu<sqy\Ì

¹

è

³y<vh}zx|vh{¥v x<sk{ z1s&sk{~S|yku<kx|yd{ x,sk{¥sqythqs8tdu³`kx1v ~s&~sqyakksk{ x<yµ1{|y

P 1

skx.sk{´x<sk&t1y8tdu s

θ

y<1kW´Æ

¿

ËZKsq

θ = 1/2

skx.KsqWz{´thy8~s`x<sk&t1y

∆t

D®C{êy<s

}|}&|}x<skuWdzhy~=z{Wd|}³ds¸u<kz}|skuC~s,g|}{ x<skuhd}s

Ω =]0, 1[

^~s,thy

h = 1/N

^¹

ÉSz{syvhz1x|vh{°~sCu<\·ku<sk{1qsav x<sk{ z1satduy<zt!skut!vy|}x|vh{½~sqy

K r

tu<sk&|sku<yZv ~sqyt1u<vhtu<sqy

Ô&ÕÖK×Ø¾Ù^Ú!ÛGF*ÝÞ à=sqkxz1sku.~sqy8y<|} z³dx|vh{1y8{ zku| z1sqy&sq

a = 0.76

b = 0.78

K r = 200

sk{ ·gd|yd{ xWhdu|sku`sqytduMd\x<u<sqy

ω

T

∆t

N

^skx

K

>~1d{1yWsqy`sk{1ysk

sqy`yz1|d{ x<y

ω = { 1, 5, 10, 22, 50 }

T ∈ { 1, 4.5, 20.125 }

∆t = { 0.1, 0.01, 0.001 }

N = { 25, 50, 100, 200 }

K = { 5, 10, 20 }

ÅÀvh{1vh|y|}uKz1sk z1sqy)wskzS¯°~s¸tdudkxu<sqyy|}{|µdx|·Ày?tdu&|Tx<vhz1y?sqyt!vy<y| sqy\Ì\

rvh&sk{ x<sku?sqyu<qyz}xdx<y ÅÍtu<qk|y|vh{¨1x<sk&t1y?~s,dkz^Ì\

RSdzS·=s\¯Sqsktx|vh{ÅÍu<sqkxd{sK~S|y< z1s}}nÌ\vh{&{1s~S|yt!vy<sthy>~=s\¯itu<sqy<y<|vh{Wd{d}±ix| z1s~sqy)v ~sqy

tu<vhtu<sqy~=z{vhzKskux

Ω ⊂ IR ²

^æ vhzu)skxxu<ssk{ESz1 u<s³Ckx1v ~s~sZyzt!skut!vy|}x|vh{W~sv ~sqy

|}·dzx.~vh{1fdkzskudzêtu<d³ sfz{1sfdttu<v¯S|}Wdx|vh{ ~sqy8tu<sk&|sku<y

K

v ~sqy8tu<vhtu<sqy&~Sz

³dt³hk|sk{~s,Ä|}u|skx

rsqk|¨t!skzx?y<sa·d|}u<s1qvh&s$s\¯it}| z1¸~1d{1ys,qvhzu<y$Æ¿ Ë sk{½zx|}}|yd{ix³.kx1v ~s$~sqykksk{ x<y

µ1{|y

Ô&ÕÖK×Ø¾Ù^Ú!ÛUT¨ÝDÞ ku|}u<sz{¥tu<vhud&sWVIXZYYVL[4\^]]:t!skuskxxd{ x ~=dttu<v 1sku$sqy ¿_ tu<sk&|sku<y

v ~sqytu<vhtu<sqyD~Sz ³dt³h\|sk{&~sÄ|u<|}1}skx«t!vhz1usqy«~skzS¯$~vhWd|}{1sqyZÅÍ³`qv Kz1vhxx<saskx³`bq1saÌ

u<sktu<qy<sk{ x<qy«y<zuT³µ1z1u<sk| ~sqy<y<vhz1yÅÀhkz{.~s>qsqy«~vhWd|}{1sqyDsky<xTz{.hy<y<sk

³ds~sÈxu|³d{sqy

u<sqkxd{sqy?|y<v qksqy|y<vhkxu|Kz1sqyC~s,chx<qy?~s,vh{z1skzu<y

n 1, 1, √

2 ^o

^Ì\

Ô&ÕÖK×Ø¾Ù^Ú!Ûed=Ý?Þ d}z1sku`³ |}x<sqyy<s½~1sqvh{ Kskusk{1qs~sqy

¿_

tu<sk&|ku<sqy`dskzu<yWt1u<vhtu<sqy~s½³

qv Kz1vhxx<sskx`~s³fbq1ssk{ ·Àvh{1kx|vh{ß~s½³xd|}}s~SzÂWd|}}³dsêÅÀvh{Âqvh{ x<u>chs\uM³¥xd|}}s~Sz

Wd|}}³ds$sk{f·gd|yd{ xhdu|skusa{1vh uskyC~s,{LSaz1~y?yzuZs vhu<~½~Sz½~vhWd|}{1s¾Ì\

$ÚÖ ÛQØÙdÖ1Ý Ñ¨stu<v ksW~s³fu<sk³dx|vh{´sk{ xus&sWyt!sqkxu<sWskx¸³·vhus~s&³°sk ud{1s

8vh{x<sk&t1y?|}{ xu|}z1¸³&kvh&8z{dzx<$y\|sk{ x<|áµ*Kz1sÑT&Kz1sqy<x|vh{½ z|¨y<sat!vys,sqyx&yz|}d{ x<s

³.~vh{{1qs¸~Sz½yt!sqkxu<sWÅÀsqyxB ~S|}u<s$~sqy

(λ _k ) _1≤k≤+∞

Ìy<zSÐxsk}s,t!vhzu~kx<sku&|}{1sku?³ ·vhus,~s

gvhzskux

Ω

ägRivh|}x)sk{8x<skusqyDt}z1yD|}WdqyZ` æ skzx vh{sk{ix<sk{1~Su<s³?·Àvhus~=z{.xd vhzu*äaÆÉdË rs {¨sqyx z¨sk{

¿hhS¿

z1s,kskxxs, z1sqyx|vh{.kx<au<qy<vh}z1sS8u\t!vh{y<s,sqyxZ{1vh{¨1Ñ¨sqyZWdx1kWdx|k|sk{y

riavhu<~vh{¨dÄ.kjÏs 1 skxlR!kjÏvh}t!skux«vh{ix«sk{8s\à=skxvh{ xu< z1s>³?qv Kz1vhxx<sskx³mbq1st!vy<y<q~sk{ x

s\¯hkx<sksk{ x?s,E#ks¸yt!sqkxu<sÆè Ë

(5)

Forme "coquotte". Forme "flèche".

oUplqp5rtsluv.slw

Æ¿

Ë$i.Lx?}³d|}u<sy î*òhþáÿdýqú¸î1ùzW÷\øMð)kù1ú.úqóôMü!óÀð{.ð³ýqòóÀðÀôî Þ qvhs æ vh}± x<sq{| z1s Þ ~S|}x|vh{

è__h

Æè

Ë.rOiavhu<~vh{¨)Ä.5js d{1~8R!5jvh}t!skux öî!ú½ñòhî1î!ôó|Sú<òhø°ó{|Súfý}|iòMü1úô~°òÏïhø\ùz 5z}g

xCsku

ç

dx¨Riv }

è

È°Å

¿hhè Ì ¿ É n ¿

ÉÒS

ÆÉBË

ç

¸hO

òhîôhîQú|Sú<òhø$ó{|Sú¸ý|SòMüú8ô~$òïøMùz

xCsku

ç

dx¨

ç

vh{ix}±½ÈhÉ°Å

¿h ÊÊÌ

¿ è ÉS

v ( x;t ) f ( x;t ) x 2 › t 2 IR v (x,t) v (x,t) xx0L W › ‰ IR ›=]0 ;L [ L

L

Ω =]0, L[

Ω ⊂ IR 2

v (x,t)

v (x,t)

Ω

x x 0

L

x ∈ Ω

t ∈ IR +

v(x, t)

f (x, t)

(I)



 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂ 2 v

∂t 2 (x, t) − ∆v(x, t) = f(x, t)

Ω × IR +∗

v(x, t) = 0

∂Ω × IR +∗

v(x, 0) = 0

Ω

∂v

∂t (x, 0) = 0

Ω.

f (x, t) = g(x) cos(ωt)

ω ∈ IR

g

L 2 (Ω)

(λ k , u k ) k∈IN ∗

Ω

λ 1 < λ 2 ≤ λ 3 ≤ · · ·

(λ k ) k∈IN ∗

+ ∞

(u k ) k∈IN ∗

L 2 (Ω)

k ∈ IN ∗

( u k ∈ H 0 1 (Ω),

− ∆u k = λ k u k .

Ω =]0, L[

(λ k , u k ) k∈IN ∗

λ k = k 2 π 2

L 2 , u k (x) = r 2

L sin µ

kπ x L

¶ .

(g k ) k∈IN ∗

g ∈ L 2 (Ω)

(u k ) k∈IN ∗

ω k = √

λ k

(I)

C 0 ([0, + ∞ [, H 1 (Ω)) ∩ C 1 ([0, + ∞ [, L 2 (Ω))

v(x, t) =

+∞

X

k=1

y k (t) u k (x).

y k (t)

IR +

IR

ν = 2π ω

ν k = ω 2π k

K ∈ IN ∗

v K (x, t) =

K

Ω ⊂ IR ²

t ∈ IR ⁺

∂ ² v

∂t ² (x, t) − ∆v(x, t) = f(x, t)

Ω × IR ^+∗

∂Ω × IR ^+∗

L ² (Ω)

(λ _k , u _k ) _k∈IN ^∗

(λ _k ) _k∈IN ^∗

(u _k ) _k∈IN ^∗

L ² (Ω)

k ∈ IN ^∗

( u _k ∈ H ₀ ¹ (Ω),

− ∆u _k = λ _k u _k .

(λ _k , u _k ) _k∈IN ^∗

λ _k = k ² π ²

L ² , u _k (x) = r 2

(g _k ) _k∈IN ^∗

g ∈ L ² (Ω)

(u _k ) _k∈IN ^∗

ω _k = √

λ _k

C ⁰ ([0, + ∞ [, H ¹ (Ω)) ∩ C ¹ ([0, + ∞ [, L ² (Ω))

y _k (t)

IR ⁺

ν = _2π ^ω

ν _k = ^ω _2π ^k

K ∈ IN ^∗

y _k (t) u _k (x).

k v( · , t) − v _K ( · , t) k L ² (Ω) ≤ 2

| g _k | ² (ω _k ² − ω ² ) ²

t∈IR ⁺ k v( · , t) − v _K ( · , t) k L ² (Ω) −→

(u _K ) _K∈IN ^∗

g ∈ H ^m (Ω)

H ⁰ (Ω) = L ² (Ω)

λ ^m _k | g k | ² < + ∞ .

g ∈ H ^m (Ω)

t∈IR ⁺

k v( · , t) − v _K ( · , t) k _L ² _(Ω) ≤ C λ ^m/2+1 _K+1 ,

(λ _k ) _k∈IN ^∗

Ω ⊂ IR ^d λ K _K→+∞ ∼ 4 π ²

mes(B _d ) mes(Ω)

B _d = ⁿ x ∈ IR ^d , | x | < 1 ^o

Ω =]0, 1[ ^d

IR ^d

N (µ) _µ→+∞ ∼ mes(B _d )

(2π) ^d mes(Ω) µ ^d/2 .

IR ^d

g ∈ H ^m (Ω)

t∈IR ⁺ k v( · , t) − v _K ( · , t) k L ² (Ω) ≤ C K ^(m+2)/d

IR ^d

IR ²

f (x, t) = χ _]a,b[ (x) cos(ω t)

χ _]a,b[