Chuong 9 - Giáo trình Matlab, BK Đà Nẵng

7/27/2019 Chuong 9 - Gio trnh Matlab, BK Nng

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CHNG 9: PHNG TRNH VI PHN O HM RING

1.KHINIMCHUNGPhngtrnhviphn ohmring(PDE)lmtlpccphngtrnh

vi phn c sbin c lp ln hn 1. Trong chng ny ta s kho st cc

phng trnh vi phn o hm ring cp 2 vi haibin c lp x v y, cdngtngqut:

+ + =

2 2 2

2 2

u u u u uA(x,y) B(x,y) C(x,y) f x,y,u, ,

x x y y x y (1)

vixoxxf,yoyyfvcc iukinbin:=o you(x, y ) b (x) =f yfu(x,y ) b (x)

=o xou(x , y) b ( y) =f xfu(x ,y) b (y) (2)CcPDE cphnthnh3loi:PDEelliptic: 2B 4AC 0

Ccphngtrnhnygnmtcchtng ngvitrngthicnbng,trngthitruynnhit,h thngdao ng

2.PHNGTRNHELLIPTICTaxtphngtrnhHelmholz:

+ = + + =

2 22

2 2

u(x, y) u(x, y)u(x,y) g(x,y) g(x,y)u(x,y) f(x,y)

x y (1)

trnmin { }= o f o fD (x,y) : x x x ,y y y vi iukinbindng:= =

= =o yo f yf

o xo f xf

u(x,y ) b (x) u(x, y ) b (x)

u(x ,y) b (y) u(x , y) b (y) (2)

Phng trnh(1) cgi lphngtrnhPoissonnug(x,y)=0vgi lphngtrnhLaplacenug(x,y)=0vf(x,y)=0. dngphngphpsai

phntachiaminthnhMx on,mi ondix=(xf xo)/MxdctheotrcxvthnhMy on,mi ondiy=(yf yo)/Mydctheotrcyvthay ohmbc2bngxpx 3 im:

+ +

j i

2i ,j 1 i ,j i ,j 1

2 2

x ,y

u 2u uu(x,y)

x xvixj=xo+jx,yj=yo+jy (3.a)


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+ +

j i

2i 1,j i ,j i 1,j

2 2

x ,y

u 2u uu(x,y)

y xviui,j=u(xj,yi) (3.b)

Nh vytimi imbntrong(xj,xi)vi1iMy 1v1jMx itnhncphngtrnhsaiphn:

+ + + ++ = =

i ,j 1 i ,j i ,j 1 i 1,j i ,j i 1,ji ,j i,j i ,j2 2

u 2u u u 2u u g u fx y

(4)

Trong :ui,j=u(xj,yi) fi,j=f(xj,yi) gi,j=g(xj,yi)

Ccphngtrnhnyspxplitheocchno thnhh phngtrnhvi

(My 1)(Mx 1)bin{ } x x y y x1,1 1,2 1,M 1 2,1 2 ,M 1 M 1,2 M 1,M 1u ,u ,...,u ,u ,...,u ,...,u ,...,u . d dngtavitliphngtrnhv iukinbindidng:

+ + = + + + +

i ,j y i ,j 1 i ,j 1 x i 1,j i 1,j xy i ,j i ,j i ,ju r (u u ) r (u u ) r (g u f ) (5a)= = = =x yi ,o xo i i ,M xf i o,j yo j M ,j yf j

u b (y ) u b (y ) u b (x ) u b (x ) (5b)

Trong :

= +

2

y 2 2

yr

2( x y )

=

+

2

x 2 2

xr

2( x y )

=

+

2 2

xy 2 2

x yr

2( x y ) (6)

Bygi takhosttipccdng iukinbin.CcbitonPDEc2 loiiukinbin: iukinbinNeumannv iukinbinDirichlet. iukin

binNeumannmtbng:

=

= o

o

xx x

u(x,y) b (y)x

(7)

Thay ohmbc1 bintri(x=xo)bngxpx 3 im: =

oi ,1 i , 1

x i

u ub (y )

2 x oi , 1 i ,1 x iu u 2b (y ) x i=1,2,...,My1 (8)

Thaythrngbucnyvo(5a) cc imbintac: + = + + + + i ,0 y i ,1 i , 1 x i 1,0 i 1,0 xy i ,0 i ,0 i ,0u r (u u ) r (u u ) r (g u f )

+ = + + + + 0y i ,1 i ,1 x i x i 1,0 i 1,0 xy i ,0 i ,0 i ,0r u u 2b (y ) x r (u u ) r (g u f )

+ = + + + 0y i ,1 x i 1,0 i 1,0 xy i ,0 i ,0 i ,0 x i2r u r (u u ) r g u f 2b (y ) x (9)

Nu iukinbintrnbindi(y=yo)cnglkiuNeumanntas vitccphngtrnhtngt vij=1,2,...,Mx1:

+ = + + + 00,j x 1,j y 0 ,j 1 0 ,j 1 xy 0,j 0 ,j 0 ,j y ju 2r u r (u u ) r g u f 2b (x ) y (10)

vbsungchogcditri(xo,yo):


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= + + +

00 y 0x 0

0,0 y 0,1 x 1,0 xy 0,0 0 ,0 0 ,0

b (x )b (y )u 2(r u r u ) r g u f 2 2

x y (11)

iukinbinDirichletchogitr hmtrnbinnncththaytrctipvophngtrnh.Tacth lygitr trungbnhcaccgitr bin lmgitr

ucaui,j.Taxydnghmpoisson() thchinthuttonny:

function[u,x,y]=poisson(f,g,bx0,bxf,by0,byf,D,Mx,My,tol,maxiter)

%giaia(u_xx+u_yy+g(x,y)u=f(x,y)

%trenmienD=[x0,xf,y0,yf]={(x,y)|x0


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forj=1:Mx

F(i,j)=f(x(j),y(i));

G(i,j)=g(x(j),y(i));

end

end

dx2=dx*dx;

dy2=dy*dy;

dxy2=2*(dx2+dy2);

rx=dx2/dxy2;

ry=dy2/dxy2;

rxy=rx*dy2;

foritr=1:maxiter

forj=2:Mx

fori=2:My

u(i,j)=ry*(u(i,j+1)+u(i,j 1))+rx*(u(i+1,j)+u(i 1,j))...

+rxy*(G(i,j)*u(i,j) F(i,j));%Pt.(5a)

end

end

ifitr>1&max(max(abs(u u0)))


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f=inline(0 ,x , y);

g=inline(0 ,x , y);

x0=0;

xf=4;

Mx=20;

y0=0;

yf=4;

My=20;

bx0=inline(exp(y) cos(y) , y);%(vd.2a)

bxf=inline(exp(y)*cos(4) exp(4)*cos(y) , y);%(vd.2b)

by0=inline(cos(x) exp(x) ,x);%(vd.3a)

byf=inline(exp(4)*cos(x) exp(x)*cos(4) ,x);%(vd.3b)

D=[x0xfy0yf];

maxiter=500;

tol=1e4;

[U,x,y]=poisson(f,g,bx0,bxf,by0,byf,D,Mx,My,tol,maxiter);

clf

mesh(x,y,U)

axis([0404 100100])

3.PHNGTRNHPARABOLIC

1. Dng phng trnh: Mt phng trnh vi phn o hm ring dngparaboliclphngtrnhmt s phnbnhit imxtithi imtcamtthanh:

=

2

2

u(x,t) u(x,t)A

x t (1)

phng trnh c th gii c ta phi cho iu kinbin u(0, t) =b0(t),=

ff xu(x , t) b (t) v iukin uu(x,0)=i0(x)

2.PhngphpEulertintngminh: pdngphngphpsaiphnhu hn, ta chia min khng gian [0, xf] thnh M on, mi on di = fx x / M vchiathigianTthnhNphn,miphnlt=T/N.Sau tathay ohmbc2 vtriv ohmbc vphica(1)bngccxpx3 imvnhn c:

++ + =

k k k k 1 ki 1 i i 1 i i

2

u 2u u u uA

x t (2)


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Cngthcnycthgignvothuttonsau,gilthuttonEul tintngminh:

++

= + + =

k 1 k k ki i 1 i 1 i 2

tu r(u u ) (1 2r)u r A

x (3)

i=1,2,...,M1

tm iukin n nhcathutton,tathaynghimth:

= ji

k k Piu e (4)

viPlsnguynkhczerovophngtrnh(3)vc:

= + + =

j j

P Pr e e (1 2r) 1 2r 1 cosP

(5)

Dotaphic||1vibitonkhngcngunnn iukin n nhl:

=

2

t 1r A

x 2

(6)

Taxydnghmfwdeuler() thchinthuttontrn

function[u,x,t]=fwdeuler(a,xf,T,it0,bx0,bxf,M,N)

%giaiau_xx=u_tvoi 0


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end

end3.PhngphpEuler li n: Ta kho st mt thut ton khc gi l thuttonEulerli, nsinhradothaythlixpx ohm ivi ohmbc1trnvphica(1):

+ + =

k k k k k 1i 1 i i 1 i i

2

u 2u u u uA

x t (7)

+

+ + = =

k k k k 1i 1 i i 1 i 2

tru (1 2r)u ru u r A

x (8)

Nu cc gi tr k0u vkMu c hai u cho trc t iu kinbin kiu

Dirichletnnphngtrnh(8) atih phngtrnh:

+ +

+ + = + + +

M M M M M M M M

L

L

k k 1 k1 1 0

k k 1

2 2k k 13 3

k k 1M 2 M 2

k k 1 kM 1 M 1 M

u u ru1 2r r 0 0 0 0

u ur 1 2r r 0 0 0

0 r 1 2r r 0 0 u u

0 0 0 1 2r r u u0 0 0 r 1 2r u u ru

(9)

iukinbinNeumann=

= 0x 0

ub (t)

xc avophngtrnhbngcch

xpx:

=

k k1 1

0

u ub (k)

2 x (10)

vghpnviphngtrnhc n k0u :

+ + =k k k k 1

1 0 1 0ru (1 2r)u ru u (11)c cphngtrnh:

+ = k k k 10 1 0 0(1 2r)u 2ru u 2rb (k) x (12)Ktqu tac ch phngtrnh:


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410

+ + +

+ =

+ + +

L

L

L

LM M M M M M M

L

L

k k0 0 0

k k 11 1

k k 12 2

k k 13 3

k k 10 M 2

k k 1 k0 M 1 M

1 2r r 0 0 0 0 u u 2rb (k) x

r 1 2r r 0 0 0 u u

0 r 1 2r r 0 0 u u

0 0 r 1 2r 0 0 u ur 0

0 0 0 0 1 2r r u u

0 0 0 0 r 1 2r u u ru

(13

iukin n nhcanghiml:

+ + =

j j

P P1

re (1 2r) re

hay: = +

1

1 2r 1 cosP

1 (14)

Taxydnghmbackeuler() thchinthuttonny:

function[u,x,t]=backeuler(a,xf,T,it0,bx0,bxf,M,N)

%Giaiau_xx=u_tvoi0


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A(i,i)=r2;%Pt.(9)

ifi>1

A(i 1,i)= r;

A(i,i 1)= r;end

end

fork=2:N+1

b=[r*u(1,k);zeros(M 3,1);r*u(M+1,k)]+u(2:M,k 1);%Pt.(9)

u(2:M,k)=trid(A,b);

end

4.PhngphpCrank Nicholson:Trong(7),xpx ohm vtrily thi imk,trongkhixpx ohm vphi. cithin,taly ohm vtriltrongbnhcaxpx ohmtihai imlkvk+1vc:

+ + + ++ + + + + =

k 1 k 1 k 1 k k k k 1 ki 1 i i 1 i 1 i i 1 i i

2 2

A u 2u u u 2u u u u

2 x x t (15)

vnhn cphngphpCrank Nicholson:+ + +

+ +

+ + = + + =

k 1 k 1 k 1 k k ki 1 i i 1 i 1 i i 1 2

tru (1 2r)u ru ru (1 2r)u ru r A

x (16)

Vi iukinbinDirichlettix0v iukinbinNeumanntixMtachphngtrnh:

+

+

+

+

+

+ + + + +

M M M M M M M

L

L

k 11

k 12

k 13

k 1M 1

k 1M

u2(1 r) r 0 0 0 0

ur 2(1 r) r 0 0 0

0 r 2(1 r) r 0 0 u

0 0 0 2(1 r) r u0 0 0 r 2(1 r) u

=

M M M M M M M

L

L

k1

k

2

k3

kM 1

kM

u2(1 r) r 0 0 0 0

ur 2(1 r) r 0 0 00 r 2(1 r) r 0 0 u

0 0 0 2(1 r) r u0 0 0 r 2(1 r) u


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[ ]

+ + +

+ +

M

k 1 k0 0

M M

r(u u )

0

0

02r b (k 1) b ( k)

(17)

iukin n nh cxc nhbng: + =

2 1 r 1 cos 2 1 r 1 cosP P

hay:

= +

1 r 1 cosP 1

1 r 1 cos P

(18)

Taxydnghmcranknicholson() thchinthuttontrn:

function[u,x,t]=cranknicholson(a,xf,T,it0,bx0,bxf,M,N)

%Giaiau_xx=u_tvoi0


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A(i,i)=r2;%Pt.(17)

ifi>1

A(i 1,i)= r;

A(i,i 1)= r;

end

end

fork=2:N+1

b=[r*u(1,k);zeros(M 3,1);r*u(M+1,k)]...

+r*(u(1:M 1,k 1)+u(3:M+1,k 1))+r1*u(2:M,k 1);

u(2:M,k)=trid(A,b);%Pt.(17)

end

giiphngtrnh: =

2

2

u(x,t) u(x,t)0 x 1, 0 t 0.1

x t (vd1)

vi iukin u:u(x,0)=sinx u(0,t)=0 u(1,t)=0 (vd2)

Nh vyvix=xf/M=1/20vt=T/N=1/100tac:

= = = 2 2

t 0.001r A 1. 0.4

x 0.05 (vd3)

Tadngchngtrnhctheat.m tmnghimca(vd1):

clearall,clc

a=1;%cacthongsocua(vd1)

it0=inline(sin(pi*x) ,x);%dieukiendau

bx0=inline(0);

bxf=inline(0);%dieukienbien

xf=1;

M=25;

T=0.1;N=100;

[u1,x,t]=fwdeuler(a,xf,T,it0,bx0,bxf,M,N);

figure(1),clf,mesh(t,x,u1)

[u2,x,t]=backeuler(a,xf,T,it0,bx0,bxf,M,N);


[u3,x,t]=cranknicholson(a,xf,T,it0,bx0,bxf,M,N);


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4.PDEparabolic2chiu:Taxtbitonphngtrnhviphn ohmringparabolichaichiumt s phnbnhit u(x,y,t):

+ =

2 2

2 2

u(x, y, t) u(x, y, t) u(x, y, t)A x y t (19)

phngtrnhcthgii ctacncho iukinbin:=

00 xu(x ,y,t) b (y,t) =

ff xu(x ,y,t) b (y,t)

=00 y

u(x,y , t) b (x,t) =ff y

u(x,y ,t) b (x,t)

v iukin uu(x,y,0)=i0(x,y)Tathay ohmbc1theot vphibngsaiphn3 imti imgia(tk+1+tk)/2nh phngphpCrank Nicholson.Tacngthaythmttrong

cc ohmbchaiuxxvuyybngxpx 3 imtithi imtkv ohmkiatitk+1vc:

++ + + + =

k k k k k k k 1 ki ,j 1 i ,j i ,j 1 i ,j 1 i ,j i ,j 1 i ,j i ,j

2 2

u 2u u u 2u u u uA

x x t (20)

Tavitphngtrnhtithi imtiptheotk+1:+ + + + ++ + + + =

k 1 k 1 k 1 k k k k 2 k 1i ,j 1 i ,j i ,j 1 i ,j 1 i ,j i ,j 1 i ,j i ,j

2 2

u 2u u u 2u u u uA

x x t (21)

Cng thc ny, c Peaceman vRachford a ra, l phng php n vtonnh phngtrnh:

( ) ( )+ + + + + + + + = + k 1 k 1 k 1 k k ky i 1,j i 1,j y i,j x i ,j 1 i ,j 1 x i,jr u u (1 2r )u r u u (1 2r )u (22a)vi0jMx 1

( ) ( )+ + + + + + + + + + + = + k 2 k 2 k 2 k 1 k 1 k 1x i ,j 1 i ,j 1 x i,j y i 1,j i 1,j y i,jr u u (1 2r )u r u u (1 2r )u (22b)vi0iMy 1

v:

=x 2

tr A

x

=

y 2t

r Ay

f 0x

x xx =M

f 0y

y yy =M

= TtN

Taxydnghmheat2D() thchinthuttonny:

function[u,x,y,t]=heat2D(a,D,T,ixy0,bxyt,Mx,My,N)

%Giaiau_t=c(u_xx+u_yy)voiD(1)


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%Dieukiendau:u(x,y,0)=ixy0(x,y)

%Dieukienbien:u(x,y,t)=bxyt(x,y,t)voi(x,y)cB

%Mx/My cacdoancodoctheotrucx/y

%N cackhoangthoigian

dx=(D(2) D(1))/Mx;

x=D(1)+[0:Mx]*dx;

dy=(D(4) D(3))/My;

y=D(3)+[0:My]*dy;

dt=T/N;

t=[0:N]*dt;

%Khoigan

forj=1:Mx+1

fori=1:My+1

u(i,j)=ixy0(x(j),y(i));

end

end

rx=a*dt/(dx*dx);

rx1=1+2*rx;

rx2=1 2*rx;

ry=a*dt/(dy*dy);

ry1=1+2*ry;

ry2=1 2*ry;

forj=1:Mx 1%Pt.(22a)

Ay(j,j)=ry1;

ifj>1

Ay(j 1,j)= ry;

Ay(j,j1)= ry;

end

end

fori=1:My 1%Pt.(22b)Ax(i,i)=rx1;

ifi>1

Ax(i 1,i)= rx;

Ax(i,i 1)= rx;

end

end


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fork=1:N

u_1=u;

t=k*dt;

fori=1:My+1%Dieukienbien

u(i,1)=feval(bxyt,x(1),y(i),t);

u(i,Mx+1)=feval(bxyt,x(Mx+1),y(i),t);

end

forj=1:Mx+1

u(1,j)=feval(bxyt,x(j),y(1),t);

u(My+1,j)=feval(bxyt,x(j),y(My+1),t);

end

ifmod(k,2)==0

fori=2:My

jj=2:Mx;

bx=[ry*u(i,1)zeros(1,Mx 3)ry*u(i,My+1)]...

+rx*(u_1(i1,jj)+u_1(i+1,jj))+rx2*u_1(i,jj);

u(i,jj)=trid(Ay,bx) ;%Pt.(22a)

end

else

forj=2:Mx

ii=2:My;

by=[rx*u(1,j);zeros(My3,1);rx*u(Mx+1,j)]...

+ry*(u_1(ii,j1)+u_1(ii,j+1))+ry2*u_1(ii,j);

u(ii,j)=trid(Ax,by);%Pt.(22b)

end

end

end

Taxtphngtrnh:

+ =

2 24

2 2

u(x, y, t) u(x, y, t) u(x, y, t)10

x y t (vd1)

trongmin:0x4,0y4vtrongkhongthiggian0t5000iukin u:

u(x,y,0)=0 (vd2a)v iukinbin:eycosx excosytix=0,x=4;y=0,y=4 (vd2b)


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Chngtrnhchngtrnhctheat2D.mdng giiphngtrnhl:

clear,clc,clf

a=1e4;

it0=inline(0 ,x , y);%(vd2a)

bxyt=inline(exp(y)*cos(x)exp(x)*cos(y) ,x , y ,t);%(vd.2b)

D=[0404];

T=5000;

Mx=40;

My=40;

N=50;

[u,x,y,t]=heat2D(a,D,T,it0,bxyt,Mx,My,N);

mesh(x,y,u)

4.PHNGTRNHHYPERBOLIC1.Dng phng trnh: Phng trnh truyn sng mt chiu l PDE dnghyperbolic:

=

2 2

2 2

u(x, t) u(x, t)A

x t (1)

0xxf,0tTiukinbin:

u(0,t)=b0(t), = ff xu(x , t) b (t)

v iubin:

u(x,0)=i0(x),=

= 0t 0

ui (x)

t

phi cchotrc phngtrnhcthgii c2. Phng php sai phn tngminh: Tng t nh khi gii PDE dngparabolic,tathay ohmbchai haivca(1)bngsaiphn3 im:

+ + + +

=

k k k k 1 k k 1i 1 i i 1 i i i

2 2

u 2u u u 2u u

A x t (2)

= fx

xM

=T

tN

vc cphngphpsaiphntngminh:

( )+ + = + + k 1 k k k k 1i i 1 i 1 i iu r u u 2(1 r)u u (3)

vi:

=

2

2

tr A

x


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V = 1i iu u(x , t) khngchotrcnntakhngthdngtrctip1iu t (3)

vik=0:

( ) + = + + 1 0 0 0 1i i 1 i 1 i iu r u u 2(1 r)u u (4)

Nh vy,taxpx iukin uv ohmbngsaiphn:

=

1 1

i i0 iu u i (x )

2 t (5)

vrtra 1iu avo(3):

( )+ = + + 1 0 0 0 1i i 1 i 1 i i 0 iu r u u 2(1 r)u u 2i (x ) t

( )+ = + + + 1 0 0 0i i 1 i 1 i 0 i

1u r u u (1 r)u i (x ) t

2 (6)

Tadng(6)cngvi iukin u c 1iu vrithayvo(3).Chl:r1 bo m n nh

chnhxccanghimtngkhirtng choxgimHplnhtllyr=1. iukin n nhcthnhn cbngcchthay(4)vo(3):

= +

1

2rcos 2(1 r)P

hay: + + =

2 2 r 1 cos 1 1 0P

Nh vy:

=

2

21 tr r A 1

x1 cos

P

Taxydnghmwave() thchinthuttontrn:

function[u,x,t]=wave(a,xf,T,it0,i1t0,bx0,bxf,M,N)

%giaiau_xx=u_ttvoi 0


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fori=1:M+1

u(i,1)=it0(x(i));

end

fork=1:N+1

u([1M+1],k)=[bx0(t(k));bxf(t(k))];

end

r=a*(dt/dx)^2;

r1=r/2;

r2=2*(1 r);

u(2:M,2)=r1*u(1:M 1,1)+(1 r)*u(2:M,1)+r1*u(3:M+1,1)...

+dt*i1t0(x(2:M));%Pt.(6)

fork=3:N+1

u(2:M,k)=r*u(1:M 1,k 1)+r2*u(2:M,k1)+r*u(3:M+1,k 1)...

u(2:M,k 2);%Pt.(3)

end

Taxtphngtrnh:

=

2 2

2 2

u(x, t) u(x, t)

x t (vd1)

0x2,0t2iukin uv iukinbin:

u(x,0)=x(1 x)=

= t 0u 0t

(vd2a)

=u(0,t) 0 u(1,t)=0 (vd2b)Tadngchngtrnhctwave.m giiphngtrnhny:

clearall,clc

a=1;

it0=inline(x.*(1x) ,x);

i1t0=inline(0);%(vd2a)bx0t=inline(0);

bxft=inline(0);%(vd2b)

xf=1;

M=20;

T=2;

N=50;


18/35

420

[u,x,t]=wave(a,xf,T,it0,i1t0,bx0t,bxft,M,N);

figure(1),clf

mesh(t,x,u)

figure(2),clf

forn=1:N%hinhdong

plot(x,u(:,n)),axis([0xf0.30.3]),pause(0.2)

end

4.PDEhyperbolic2chiu:PhngtrnhtruynsnghaichiulPDEdnghyperbolic:

+ =

2 2 2

2 2 2

u(x, y, t) u(x, y, t) u(x, y, t)A

x y t (8)

0xxf,0yyf,0tTiukinbin:=

0xu(0,y,t) b (y,t) =

ff xu(x ,y,t) b (y,t)

=0y

u(x,0,t) b (x,t) =ff y

u(x,y ,t) b (x,t)

v iubin:

= 0u(x,y,0) i (x,y)=

=

0t 0

u(x,y)i (x,y)

t

Tngt nh hmmtbin,tathay ohmbc2bngxpx 3 im:+

+ + + + ++ =

k k k k k k k 1 k k 1

i ,j 1 i ,j i ,j 1 i 1,j i ,j i 1,j i ,j i ,j i ,j2 2 2

u 2u u u 2u u u 2u uAx y t

(9)

= fx

xx

M = f

y

yy

M =

Tt

N

vnhn i nphngphptngminh:

( ) ( )+ + + = + + + + k 1 k k k k k k 1i ,j x i ,j 1 i ,j 1 x y i ,j y i 1,j i 1,j i ,ju r u u 2(1 r r )u r u u u (10)vi:

=

2

x 2

tr A x

=

2

y 2

tr A y

V = 1i,j i iu u(x ,y , t) khngchotrcnntakhngthdngtrctip1i ,ju t

(10)vik=0:

( ) ( ) + + = + + + + 1 0 0 0 0 0 1i ,j x i ,j 1 i ,j 1 x y i ,j y i 1,j i 1,j i ,ju r u u 2(1 r r )u r u u u (11)Nh vy,taxpx iukin uv ohmbngsaiphn:


19/35

421

=

1 1i ,j i ,j

0 j i

u ui (x ,y )

2 t (12)

vrtra 1i ,ju avo(11):

( ) ( )+ + = + + + + +

1 0 0 0 0i ,j x i ,j 1 i ,j 1 y i 1,j i 1,j

0x y i ,j 0 j i

1u r u u r u u

22(1 r r )u i (x ,y ) t (13)

iukin n nh:

= +

2

2 2

4A tr 1

x y

Taxydnghmwave2D() thchinthuttontrn:

function[u,x,y,t]=wave2D(a,D,T,it0,i1t0,bxyt,Mx,My,N)

%giai

a(u_xx

+u_yy)

=u_tt

voi

D(1)


20/35

422

rxy1=1 rx ry;

rxy2=rxy1*2;

u_1=u;

fork=0:N

t=k*dt;

fori=1:My+1%dieukienbien

u(i,[1Mx+1])=[bxyt(x(1),y(i),t) bxyt(x(Mx+1),y(i),t)];

end

forj=1:Mx+1

u([1My+1],j)=[bxyt(x(j),y(1),t);bxyt(x(j),y(My+1),t)];

end

ifk==0

fori=2:My

forj=2:Mx%Pt.(13)

u(i,j)=0.5*(rx*(u_1(i,j 1)+u_1(i,j+1))...

+ry*(u_1(i 1,j)+u_1(i+1,j)))+rxy1*u(i,j)+dt*ut(i,j);

end

end

else

fori=2:My

forj=2:Mx

u(i,j)=rx*(u_1(i,j 1)+u_1(i,j+1))...

+ry*(u_1(i 1,j)+u_1(i+1,j))+rxy2*u(i,j) u_2(i,j);

end

end

end

u_2=u_1;

u_1=u;

mesh(x,y,u),axis([0202 .1.1])

pause(0.1);end

Taxtphngtrnh:

+ =

2 2 2

2 2 2

u(x, y, t) u(x, y, t)1 u(x,t)

4 x y t (vd1)

0x2,0y2,0t2


21/35

423

iukin uv iukinbin:u(0,y,t)=0 u(2,y,t)=0 u(x,0,t)=0 u(x,2,t)=0 (vd2)

=

yxu(x,y,0) 0.1sin sin

2 2

=

=

t 0

u0

t (vd3)

Tadngchngtrnhctwave2D.m giiphngtrnhny:

clearall,clc

it0=inline(0.1*sin(pi*x)*sin(pi*y/2) ,x , y);%(vd3)

i1t0=inline(0 ,x , y);

bxyt=inline(0 ,x , y ,t);%(vd2)

a=.25;

D=[0202];

T=2;

Mx=40;

My=40;

N=40;

[u,x,y,t]=wave2D(a,D,T,it0,i1t0,bxyt,Mx,My,N);

5.PHNGPHPPHNTHUHN(FEM)GIIPDEPhngphpFEMdng tmnghimscaPDEvi iukinbin.

TaxtmtPDEdngelliptic:

+ + =

2 2

2 2

u(x, y) u(x, y)g(x,y)u(x,y) f(x,y)

x y (1)

trongminDbaobibinBvtrnbinccc iukin:u(x,y)=b(x,y) trnB (2)

CcbcdngFEM giiphngtrnhgm:)ChiaminDthnhNsmincon{S1,S2,...,SNs}cdnghnhtamgic)Mt v trcaNnntv nhschngbt ut ccnttrnbin:n=1,2,...,Nbvccntbntrong:n=Nb+1,Nb+2,...,Nn)Xc nhcchmnisuy,hnhdngvc s:

{ } = = n n ,s s(x,y) s 1,...,N (x,y) D (3a) = + +n,s n ,s n,s n,s(x, y) p (1) p (2)x p (3)y

chomiminSs (3b)


22/35

424

ivittccccmincons=1:Nsvccntn=1:Nnsaochonbng1ch ntnvbngzeroticcntkhc.Lc nghimxpx caPDElthptuyntnhcacchmc s n(x,y):

[ ] [ ]=

= = nN

T

n ni 1

u(x,y) c (x,y) c (x,y)

= = +

= + b n

b

N N

n n n ni 1 i N 1

c (x,y) c (x,y)

[ ] [ ] [ ] [ ]= + T T

1 1 2 2c c (4)

Trong :

[ ] = L bT

1 1 2 N [ ] = L bT

1 1 2 Nc c c c (5a)

[ ] + + = Lb b nT

2 N 1 N 2 N [ ] + + = Lb b nT

2 N 1 N 2 Nc c c c (5b)

Vimimincon,nghimnycthvitdidng:

= =

= = + + n nN N

s n n ,s n n,s n,s n ,si 1 i 1

(x,y) c (x,y) c p (1) p (x) p (3)y (6)

) tccgitr cah sntbintrong[c1]bngccgitrbintngngvi iukinbin)Xc nhtr scah sntbntrongtrong[c2]bngcchgiihphngtrnh:

[A2][c2]=[d] (7)

trong :

[ ]

=

= + sN

TT

1 2,s 1,s 2 ,s 1,s s

s 1

A Sx x y y

{ }=

sN

T

s s 2 ,s 1,s s

s 1

g(x , y ) S (8)

= L bT

1,s 1,s 2 ,s N ,s

= L

b

T1,s 1,s 2 ,s N ,sp (2) p (2) p (2)

x

=

Lb

T

1,s 1,s 2 ,s N ,sp (3) p (3) p (3)y

[ ]

=

= + sN

TT

2 2,s 2 ,s 2,s 2 ,s s

s 1

A Sx x y y


23/35

425

{ }=

sN

T

s s 2,s 2 ,s s

s 1

g(x , y ) S (9)

+ + = Lb b nT

2,s N 1,s N 2,s N ,s

+ +

=

L

b b n

T

2,s N 1,s N 2,s N ,sp (2) p (2) p (2)x

+ +

=

Lb b n

T

2,s N 1,s N 2,s N ,sp (3) p (3) p (3)y

[ ] [ ][ ]=

= sN

1 1 s s 2 ,ss 1

d A c f(x ,y ) S (10)

(xs,ys)ltrongtmcaminconSsFEMdatrnnguyntclnghimca(1)cthnhn cbngcch

cctiuhohm:

= + 22

R

J u(x,y) u(x,y)x y

} +2g(x,y)u (x,y) 2f(x,y)u(x,y) dxdy (11)Viu(x,y)=[c]T[(x,y)]tac:

[ ] [ ] [ ][ ]{ [ ] [ ] [ ] [ ] = + T T TR

J c c c cx x y y

[ ] [ ][ ] [ ] [ ] [ ]} + T T T

g(x,y) c c 2f(x, y) c dxdy (12)iukin hmnycctiutheo[c]l:

[ ][ ] [ ] [ ]{ [ ] [ ] = + T T22

R

dJ c c

d c x x y

[ ][ ] [ ] [ ]} + T2 2g(x,y) c 2f(x,y) dxdy = 0 (13)

[ ][ ] [ ][ ]=

+ + = sN

1 1 2 2 s s 2 ,ss 1

A c A c f(x ,y ) S 0 (14)

xydnghmc s n,s(x,y) th s ivimintn=1,2,...,Nnvmimincons=1,2,...,Nstaxydnghmfembasisftn():

functionp=fembasisftn(N,S)

%p(i,s,1:3):cachesocuamoihamcosophi(i)

%cuamientamgiac(miencon)thus

%N(n,1:2):x&ytoadocuanutthun


24/35

426

%S(s,1:3):nutthuscuamiencontamgiacthus

Nn=size(N,1);%tongsonut

Ns=size(S,1);%tongsocacnutcuamiencontamgiac

forn=1:Nn

fors=1:Ns

fori=1:3

A(i,1:3)=[1N(S(s,i),1:2)];

b(i)=(S(s,i)==n);%hamcosothunbang1chionutthun

end

pnt=A\b ;

fori=1:3\

p(n,s,i)=pnt(i);

end

end

end

xc nhccvect h s[c]canghim(4)nh (7)vcc athcnghim s(x,y) nh (6) ivimimincons=1,2,...,Nstaxydnghmfemcoef():

function[U,c]=femcoef(f,g,p,c,N,S,Ni)

%p(i,s,1:3):cachesocuahamcosophi(i)cuamienconthun

%c=[.11.00.]voicacgiatribienva0voicacnutbentrong

%N(n,1:2):x&ytoajdocuanutthun

%S(s,1:3):nutthuscuamienconthus

%Ni:sonutbentrong

%U(s,1:3):cachesocuap1+p2(s)x+p3(s)ycuamoimiencon

Nn=size(N,1);%tongsonutbangNb+Ni

Ns=size(S,1);%tongsomiencon

d=zeros(Ni,1);

Nb=Nn Ni;

fori=Nb+1:Nn

forn=1:N_n

fors=1:Ns

xy=(N(S(s,1),:)+N(S(s,2),:)+N(S(s,3),:))/3;%trongtam

%phi(i,x)*phi(n,x)+phi(i,y)*phi(n,y) g(x,y)*phi(i)*phi(n)


25/35

427

pvctr=[p([in],s,1)p([in],s,2)p([in],s,3)];

tmpg(s)=sum(p(i,s,2:3).*p(n,s,2:3))...

g(xy(1),xy(2))*pvctr(1,:)*[1xy]*pvctr(2,:)*[1xy] ;

dS(s)=det([N(S(s,1),:)1;N(S(s,2),:)1;N(S(s,3),:)1])/2;

%dientichmiencon

ifn==1

tmpf(s)= f(xy(1),xy(2))*pvctr(1,:)*[1xy] ;

end

end

A12(i Nb,n)=tmpg*abs(dS) ;%Pt.(8),(9)

end

d(iNb)=tmpf*abs(dS) ;%Pt.(10)

end

d=d A12(1:Ni,1:Nb)*c(1:Nb) ;%Pt(10)

c(Nb+1:Nn)=A12(1:Ni,Nb+1:Nn)\d;%Pt.(7)

fors=1:Ns

forj=1:3

U(s,j)=c*p(:,s,j);%Pt.(6)

end

end

TrckhidngFEM giimtPDEtaxemth hmc s(hmhnhdng)n(x,y) ivimintn=1,2,...,Nn c nhngha ivittc ccminconhnhtamgicsaochonbng1ch tintnvbng0ticcntkhcc tobi hmfembasisfth() hot ng nh th no. Ta s v hm hnhdngcamin cchiathnh4tamgicnh hnhsau:

To cant: SntmiminconN=[ 1 1; S=[1 2 5;

1 1; 2 3 5;1 1; 3 4 5;

1 1; 1 4 5];0 0.5];

1 10

1

0

n=4

n=5

n=3

n=2n=1 S1

S4

S3

S2


26/35

428

theo hai cch. Trc ht ta to cc hm c s bng cch dng hmfembasisftn() v v mt hm trong s bng cch dng lnh MATLABmesh()nh hnha.Th hai,khngtorahmc s,tadnglnhMATLABtrimesh() v cchmhnhdngchoccntn=2,3,4v5nh hnhbe.Hnhfl th cathptuyntnhcacchmc s:

[ ] [ ]=

= = nN

T

n nn 1

u(x,y) c (x,y) c (x, y) (15)

ctr scntimintn.Tachychngtrnhctshowbasic.m:

clearall,clc

N=[11;11;1 1;1 1;0.20.5];%danhsachcacnuttrenhinh1

Nn=size(N,1);%sonut

S=[125;235;345;145];%danhsachcamiencontrenhinh1

Ns=size(S,1);%somiencon

figure(1),clf

fors=1:Ns

nodes=[S(s,:)S(s,1)];

fori=1:3

plot([N(nodes(i),1)N(nodes(i+1),1)],...

[N(nodes(i),2) N(nodes(i+1),2)])

holdon

end

end

ins=[12345];%danhsachcacnutmacachamcosoduocve

foritr=1:5

in=ins(itr);

ifitr==1

fori=1:length(xi)

forj=1:length(yi)

Z(j,i)=0;fors=1:Ns

ifinpolygon(xi(i),yi(j),N(S(s,:),1),N(S(s,:),2))>0

Z(j,i)=p(in,s,1)+p(in,s,2)*xi(i)+p(in,s,3)*yi(j);

break;

end

end


27/35

429

end

end

subplot(321),mesh(xi,yi,Z)%hamcosocuanut1

else

c1=zeros(size(c));

c1(in)=1;

subplot(320+itr)

trimesh(S,N(:,1),N(:,2),c1)%hamcosocuacacnut25

end

end

c=[01230];%cacgiatritaicacnut

subplot(326)

trimesh(S,N(:,1),N(:,2),c)%hamtonghopohinhf

-10

1

-10

10

0.5

1

-10

1

-10

10

0.5

1

-10 1

-101

0

0.5

1

-10 1

-101

0

0.5

1

-10

1

-10

10

0.5

1

-10

1

-10

10

2

4

c=[01230];

p=fembasisftn(N,

S);

x0= 1;

xf=1;

y0= 1;

yf=1;%cacmien

figure(2),clf

Mx=50;


28/35

430

My=50;

dx=(xf x0)/Mx;

dy=(yfy0)/My;

xi=x0+[0:Mx]*dx;

yi=y0+[0:My]*dy;

Vd:GiiphngtrnhLaplace:

= + =

2 22

2 2

u(x, y) u(x, y)u(x,y) f(x,y)

x y (1)

trnmin 1 x 1; 1 y 1vi:

1f(x,y)= +1

0

(2)

v iukinbinlu(x,y)=0timi imtrnbin.

vi(x,y)=(0.5,0.5)vi(x,y)=(0.5, 0.5)

ccch khc


29/35

431

giibitonnybngFEM,taxc nh12 imtrnbinv19 imbntrong, nhschngvchiaminch nhtthnh36minconhnhtamgicnh hnhv trn.Tiptheotaxydngchngtrnhctlaplace.m giibiton

clearall,clc

N=[10;1 1;1/2 1;0 1;1/2 1;1 1;10;11;1/21;01;

1/21;11; 1/2 1/4; 5/8 7/16;3/4 5/8;1/2 5/8;

1/4 5/8;3/8 7/16;00;1/21/4;5/87/16;3/45/8;

1/25/8;1/45/8;3/87/16;9/16 17/32;7/16 17/32;

1/2 7/16;9/1617/32;7/1617/32;1/27/16];%nut

Nb=12;%sonuttrenbien

S=[11112;11119;101119;4519;5719;567;1215;2315;

31517;3417;41719;131719;11319;11315;7822;8922;

92224;91024;101924;192024;71920;72022;131418;

141516;161718;202125;212223;232425;142628;

162627;182728;212931;232930;253031;

262728;293031];%miencontamgiac

fexemp= (norm([xy]+[0.50.5])


30/35

432

Mx=16;

dx=(xf x0)/Mx;

xi=x0+[0:Mx]*dx;

My=16;

dy=(yfy0)/My;

yi=y0+[0:My]*dy;

fori=1:length(xi)

forj=1:length(yi)

fors=1:Ns

ifinpolygon(xi(i),yi(j),N(S(s,:),1),N(S(s,:),2))>0

Z(i,j)=U(s,:)*[1xi(i)yi(j)] ;%Pt.(4.5b)

break;

end

end

end

end

figure(2);

clf;

mesh(xi,yi,Z)

%desosanh

bx0=inline(0);

bxf=inline(0);

by0=inline(0);

byf=inline(0);

D=[x0xfy0yf];

[U,x,y]=poisson(f,g,bx0,bxf,by0,byf,D,Mx,My,1e6,50);

figure(3)

clf;

mesh(x,y,U)

6.GUICAMATLAB GIIPDE1.Ccphngtrnhcthgii cbngPDETOOL:Cngc PDETOOLcaMATLABcthdng giiccloiphngtrnhsau:

a.Phngtrnhelliptic:Tas giiphngtrnhelliptic + =(c u) au f (1)

vi iukinbn:


31/35

433

=

rhu r Dirichlet

nc u+qu=g Neumann (2)

trnbin,trong rn lvect phptuyn.

Trongtrnghpul ilngvhng,phngtrnh(1)tr thnh:

+ + =

2 2

2 2u(x, y) u(x, y)c au(x,y) f(x,y)x y (3)

vnu iukinbin iviphnbinbntril iukinbinNeumann

dng=

=

00

xx x

u(x,y)b (y)

xth(2)cthvitthnh:

+ +

= + =

r r rx x y

u(x, y) u(x, y)e c e e qu(x,y)

x y

u(x,y)

c qu(x,y) g(x,y)x

(4)

vvect phptuyncabinphil =r r

xn e b.Phngtrnhparabolic:Taxgiiphngtrnh:

+ + =

u

(c u) au d ft

(5)

trnminvtrongkhongthigian0tT,vi iukinbnging(2)viukin uu(t0)

c.Phngtrnhhyperbolic:

+ + =

2

2u(c u) au d f

t (6)

trnminvtrongkhongthigian0tT,vi iukinbnging(2)viukin uu(t0),u(t0)

d.Phngtrnhgitr ring:

+ = u

(c u) aut

(7)

trnminvimtgitr ringchabitv iukinbintngt (2).

Cngc PDETOOLcngcthdng giih phngtrnhdng: + + = + + =

11 1 12 2 11 1 12 2 1

21 1 22 2 21 1 22 2 2

(c u ) (c u ) a u a u f

(c u ) (c u ) a u a u f (8)

trnminvi iukinbinDirichlet:

=

1 111 12

2 221 22

u rh h

u rh h (9)

hay iukinNeumanntngqut:


32/35

434

+ + + =

+ + + =

r r

r r11 1 12 2 11 1 12 2 1

21 1 22 2 21 1 22 2 2

n(c u ) n(c u ) q u q u g

n(c u ) n(c u ) q u q u g (10)

hay iukinbinhnhp:

=

11 12

21 22

c cc

c c

=

11 12

21 22

a aa

a a

=

1

2

ff

f

=

1

2

uu

u

=

11 12

21 22

h hh

h h

=

1

2

rr

r

=

11 12

21 22

q qq

q q

=

1

2

gg

g

2.S dngPDETOOL:PDETOOLgiiphngtrnhviphn ohmringbng cch dng phng php FEM. gii phng trnh ta theo ccbcsau:

) Nhp lnhpdetool vo ca s lnh MATLAB. Ca s PDE toolbox

xuthin.Tacthbt/tttu chnGridbngcchbmvoGridtrnmenuOption.TacngcthhiuchnhphmvitrcxvybngcchchnAxesLimittrongnemuOption

Numunchocchnhgnvoli,tachnSnaptrongmenuOption.Numunt l xchcatrcxvtbngnhau hnhtrnnhnkhngginghnhelliptachnAxesEqualtrongmenuOption.) v min ta dng menuDraw hay cc icon trn thanh cng cngayphadiccmenu.) t iukinbintadngmenuBoundaryhayicon.Tabmlntng onbin t iukinchon.


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435

) TiptheotatolibngcchdngmenuMeshhayicon. tinhchnhlitabmvoRefineMeshhayicon) Tip theo ta m t dng phng trnh v cc thng s ca nbngcchdngmenuPDE.Mun th,tam menuPDEhaychn iconPDEvchnPDESpecificationvchoccthamscaphngtrnh.) giiphngtrnhtadngmenuSolvehaychnicon= .Tachnmenu con Parameters nhp iu kin u v khong thi gian tmnghim)Numunv ktqu,tadngmenuPlot

3.Mtsvd:a.Vd 1:GiiphngtrnhLaplace:

= + =

2 22

2 2

u(x, y) u(x, y)

u(x,y) 0x y (vd1.1)trongmin0x4,0y4vicc iukinbin:

u(0,y)=ey cosy u(4,y)=eycos4 e4cosy (vd1.2)u(x,0)=cosx ex u(x,4)=e4cosx excos4 (vd1.3)giiphngtrnhtathchinccbcsau:M cngc PDETOOL.VomenuOption|AxesLimithiuchnhli phm vi gi tr ca x v y l [0 5] ri chnApply vClose. ChnOption|AxesEqualBmvoicon v hnhvung.Khiv xong,nucha ngkchthctabm pvo itngbygi ctnlR1 hiuchnhlithnhLeft:0,Bottom:0,Height:4,Width:4.Bmvoiconth ngbinca itngcmu .Trnmionbintacho iukinbintheo(vd1.2)v(vd1.3). ghi iukin

bincho onnotabm pchutln on . iukinbin chol iukinbinDirrichlet.Trnbintri,taghi iukinbin:

h=1,r=exp(y) cos(y)

trnbinphi:h=1,r=eycos4 e4cosy

trnbindi:h=1,r=cosx ex

vtrnbintrn:h=1,r=e4cosx excos4


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436

Bm pchutvoiconPDEvchnphngtrnhdngellipticvccthngstheo(vd1.1):c=1,a=0,f=0Bm pchutvoicon tolivsau tinhchnhn.Bm pchutvoicon= giiphngtrnh.VomenuPlot|Parameters chncchv vsau v raktqu

b.Vd 2:Giiphngtrnhparabolic:

+ =

2 24

2 2

u(x, y, t) u(x, y, t) u(x, y, t)10

x y t (vd2.1)

trongmin0x4,0y4v0t5000 vicc iukin uv iubin:

u(x,y,0)=0 (vd2.2a)

u(x,y,t)=ey

cosx ex

cosyvix=0,x=4,y=0,y=4 (vd2.2b)giiphngtrnhtatheoccbcsau:M cngc PDETOOL.VomenuOption|AxesLimithiuchnhli phm vi gi tr ca x v y l [0 4] ri chnApply vClose. ChnOption|AxesEqualBmvoicon v hnhvung.Khiv xong,nucha ngkchthctabm pvo itngbygi ctnlR1 hiuchnhlithnhLeft:0,Bottom:0,Height:4,Width:4.Bmvoiconth ngbinca itngcmu .Trnmionbintacho iukinbintheo(vd2.2b). ghi iukinbinchoonnotabm pchutln on . iukinbin chol iukinbinDirrichlet.Trnbintri,taghi iukinbin:

h=1,r=exp(y) cos(y)trnbinphi:

h=1,r=eycos4 e4cosytrnbindi:

h=1,r=cosx exvtrnbintrn:

h=1,r=e4cosx excos4Bm pchutvo iconPDEvchnphngtrnhdngparabolicv cc thng s theo (vd2.1): c = 1e4, a = 0, f = 0, d = 1. Trong menuSolve|ParameterstaghiTime:0:100:5000,u(t0)=0(iukin u).Bm pchutvoicon tolivsau tinhchnhn.Bm pchutvoicon= giiphngtrnh.


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VomenuPlot|Parameters chncchv vsau v raktqu

c.Vd 3:Giiphngtrnhhyperbolic:

+ =

2 2 2

2 2 2

u(x, y, t) u(x, y, t) u(x, y, t)1

4 x y t

(vd3.1)

trongmin0x2,0y2v0t2 vicc iukinbinzerov iukin u:u(0,y,t)=0 u(2,y,t)=0 u(x,0,t)=0 u(0,2,t)=0 (vd3.2)u(x,y,0)=0.1sin(x)sin(y/2) u/t(x,y,0)=0vit=0 (vd3.3)giiphngtrnhtatheoccbcsau:

M cngc PDETOOL.VomenuOption|AxesLimithiuchnhli phm vi gi tr ca x v y l [0 2] ri chnApply vClose. Chn

Option|AxesEqualBmvoicon v hnhvung.Khiv xong,nucha ngkchthctabm pvo itngbygi ctnlR1 hiuchnhlithnhLeft:0,Bottom:0,Height:2,Width:2.Bmvoiconth ngbinca itngcmu .Trnmionbintacho iukinbintheo(vd3.2). ghi iukinbinchoonnotabm pchutln on . iukinbin chol iukinbinDirrichlet.Trnbintri,taghi iukinbin:

h=1,r=0trnbinphi:

h=1,r=0trnbindi:

h=1,r=0vtrnbintrn:

h=1,r=0Bm pchutvo iconPDEvchnphngtrnhdngparabolicvccthngstheo(vd2.1):c=1/4,a=0,f=0,d=1.TrongmenuSolve|ParameterstaghiTime:0:0.1:2,u(t0)=0.1*sin(pi*x).*sin(pi*y/2).Bm pchutvoicon tolivsau tinhchnhn.Bm pchutvoicon= giiphngtrnh.VomenuPlot|Parameters chncchv vsau v raktqu

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Chuong 9 - Giáo trình Matlab, BK Đà Nẵng