23復 傾僯儊乕僔儑儞


偙傟傑偱偺妛廗傪傕偲偵丄偝傑偞傑側傾僯儊乕僔儑儞傪嶌惉偟偰傒傑偟傚偆丅

1. 攇偺揱攄

>with(plots):

>animate(sin(x-t),x=0..4*Pi,t=0..10)


偲擖椡偟丄昤偐傟偨僌儔僼傪僋儕僢僋偟偰昞帵偝傟傞儊僯儏乕僶乕忋偺傾僯儊乕僔儑儞嵞惗儃僞儞傪 僋儕僢僋偡傞偲丄傾僯儊乕僔儑儞偑巒傑傝傑偡丅儊僯儏乕僶乕忋偺儃僞儞傪巊偭偰丄傾僯儊乕僔儑儞偺幚峴懍搙傪曄偊偨傝 楢懕嵞惗儌乕僪偵愝掕偡傞偙偲側偳偑偱偒傑偡丅

拲堄 u(x,t)=sin(x-t)偼旝暘曽掱幃utt=uxx(攇摦曽掱幃)傪枮偨偟傑偡丅 偡側傢偪攇摦曽掱幃偺夝偱偡丅

丂恑峴攇夝偲傛偽傟傞旝暘曽掱幃偺夝偺僔儏儈儗乕僔儑儞傪峴偭偨傝丄傜偣傫傪堦掕偺 妏懍搙偱夞揮偝偣偰傒傑偟傚偆丅

>animate(tanh(x-t),x=-20..20,t=0..25)
(恑峴攇夝偺僔儈儏儗乕僔儑儞)


>animate([r*cos(r-t),r*sin(r-t),r=0..6*Pi],t=0..10,numpoints=100,scaling=constrained)
乮夞揮偡傞傜偣傫偺僔儈儏儗乕僔儑儞乯




拲堄
(1)u(x,t)=tanh(x-t)偼旝暘曽掱幃ut丂uxx+(2u-1)(1-u2)傪枮偨偟傑偡丅 偙偺傛偆偵丄偁傞娭悢冇(y)偲掕悢c傪梡偄偰u(x,t)=冇(x-ct)偲昞偣傞堦掕攇宍偱堦掕懍搙c偱恑傓夝傪恑峴攇夝偲偄偄傑偡丅 悢棟惗暔妛傗擱從棟榑傊偺墳梡忋傕廳梫側夝偱偡丅
(2)嬋慄(rcosr,rsinr)丄r伕[a,b]偼傾儖僉儊僨僗偺傜偣傫偲傛偽傟傑偡丅 椺戣偺夞揮偡傞傜偣傫偼丄寢徎惉挿傪婰弎偡傞悢棟儌僨儖偵尰傟傞夝偱偡丅

Kdv(Korteweg-de Iries) 曽掱幃丂ut+6uux+uxxx=0偺僜儕僩儞夝(屒棫攇夝)
u(x,t)=2k2sech2(kx-4k3t)

(k偼掕悢)傗俀僜儕僩儞夝
u(x,t)=2(log(1+ce2kx-8k3t+de2hx-8h3t+cd(k-h)2e2kx-8k3t+2hx-8h3t/(k+h)2))xx

(c,d,k,h偼掕悢)偺僔儈儏儗乕僔儑儞傪峴偭偰傒傑偟傚偆丅

>animate(2*sech(x-4*t)2,x=-25..25,t=-5..5 ,numpoints=200)
乮僜儕僩儞夝(k=1偺応崌乯偺僔儈儏儗乕僔儑儞)


>v:=(x,t)->1+exp(2*x-8*t)+exp(x-t)+9*exp(2*x-8*t+2*x-t)/16

>u:=(x,t)->2*diff(log(v(x,t)),x,x)

>animate(u(x,t),x=-30..30,t=-6..6,numpoints=200)
(2僜儕僩儞夝(c=d=k=1,h=1/2偺応崌)偺僔儈儏儗乕僔儑儞乯




2. 僒僀僋儘僀僪
僒僀僋儘僀僪偼丄墌偑 x 幉偵愙偟側偑傜夞揮偡傞偲偒丄偦偺廃忋偵屌掕偝傟偨揰 P 偺婳愓偲偟偰掕媊偝傟傑偡丅 墌偺敿宎偲 a 偲偡傞偲丄

x = a (兤-sin 兤)
y = a (1-cos 兤)

偲昞偣傑偡丅a=1 偺応崌偵丄兤傪 0 偐傜 2兾 傑偱摦偐偟偨帪偺婳愓傪昤偄偰傒傑偟傚偆丅


>with(plots):

>animatecurve([t-sin(t), 1-cos(t), t = 0 .. 2*Pi], scaling = constrained, frames = 16)
(僒僀僋儘僀僪傪昤偔乯


>animate([cos(s)+t, sin(s)+1, s = 0 .. 2*Pi], t = 0 .. 2*Pi, scaling = constrained)
(夞揮偡傞墌偺僔儈儏儗乕僔儑儞乯


>animate([s*(t-sin(t))+(1-s)*t, s*(1-cos(t))+1-s, s = 0 .. 1], t = 0 .. 2*Pi, scaling = constrained)
(摦宎偺僔儈儏儗乕僔儑儞乯


>display({%, %%, %%%})




3. 怳傝巕偺塣摦
尨揰 (0,0) 偵夞揮幉傪傕偮怳傝巕偑丄 廳椡偺嶌梡偵傛傝塣摦偡傞條巕傪峫偊傑偡丅
帪崗 t 偵偍偄偰丄怳傝巕偺巺偑墧捈壓岦偒偲敿帪寁夞傝傪惓偺岦偒偵偲偭偰丄s(t) 偺妏搙傪側偡帪

ml s''(t) = -mg sin s(t)

偑惉傝棫偪傑偡丅偙偙偱丄m 偼偍傕傝偺幙検丄g 偼廳椡壛懍搙丄l 偼巺偺挿偝傪昞偟傑偡丅 l=g 偺応崌偵丄弶婜忦審 s(0)=0丄s'(0)=0.8 傪枮偨偡夝傪媮傔丄僔儈儏儗乕僔儑儞偟偰傒傑偟傚偆丅 尩枾夝偼媮傔傜傟側偄偺偱丄dsolve 僐儅儞僪偺僆僾僔儑儞偱 numeric 偲巜掕偟偰丄悢抣夝傪媮傔傞偙偲偵偟傑偡丅

>with(plots):

>dsolve({diff(s(t), t, t) = -sin(s(t)), s(0) = 0, (D(s))(0) = .8}, numeric, output = listprocedure);

>odeplot(%, [sin(s(t)), -cos(s(t))], t = 0 .. 30, style = point, frames = 100, scaling = constrained)


幚廗23.1丂 忋偺怳傝巕偺塣摦偵偮偄偰丄弶婜忦審偑埲壓偺応崌偵偮偄偰偺傾僯儊乕僔儑儞傪嶌惉偟偰傒傑偟傚偆丅
丂丂(1) s(0)=兾/2丄s'(0)=0
丂丂(2) s(0)=0丄s'(0)=1.8
丂丂(3) s(0)=0丄s'(0)=2.2

[惓夝椺]