Example 6.1 Rev 1.html

Example 6.1 Rev 1.mw

Example 6.1

(1)

> ge:=diff(u(x,y),y$2)=-epsilon^2*diff(u(x,y),x$2);

(2)

> bc1:=u(x,y)-0;

(3)

> bc2:=u(x,y)-0;

(4)

> bc3:=u(x,y)-0;

(5)

> bc4:=u(x,y)-1;

(6)

> epsilon:=1;

(7)

> dydxf:=1/2/h*(-u[m+2](zeta)-3*u[m](zeta)+4*u[m+1](zeta));

(8)

> dydxb:=1/2/h*(u[m-2](zeta)+3*u[m](zeta)-4*u[m-1](zeta));

(9)

> dydx:=1/2/h*(u[m+1](zeta)-u[m-1](zeta));

(10)

> d2ydx2:=1/h^2*(u[m-1](zeta)-2*u[m](zeta)+u[m+1](zeta));

(11)

> bc1:=subs(diff(u(x,y),x)=subs(m=0,dydxf),u(x,y)=u[0](zeta),x=0,bc1);

(12)

> bc2:=subs(diff(u(x,y),x)=subs(m=N+1,dydxb),u(x,y)=u[N+1](zeta),x=1,bc2);

(13)

> N:=10;

(14)

> eq[0]:=bc1;

(15)

> eq[N+1]:=bc2;

(16)

> for i from 1 to N do eq[N+1+i]:=diff(u[N+1+i](zeta),zeta)= subs(diff(u(x,y),x$2) = subs(m=i,d2ydx2),diff(u(x,y),x) = subs(m=i,dydx),u(x,y)=u[i](zeta),x=i*h,rhs(h^2/epsilon^2*ge));od;

(17)

> u[0](zeta):=(solve(eq[0],u[0](zeta)));

(18)

> u[N+1](zeta):=solve(eq[N+1],u[N+1](zeta));

(19)

> for i from 1 to N do eq[i]:=diff(u[i](zeta),zeta)=u[N+1+i](zeta);od;

(20)

> for i from 1 to N do eq[i]:=eval(eq[i]);od;for i from 1 to N do eq[N+1+i]:=eval(eq[N+1+i]);od;

(21)

> eqns:=[seq(rhs(eq[j]),j=1..N),seq(rhs(eq[N+1+j]),j=1..N)];

(22)

> Y:=[seq(u[i](zeta),i=1..N),seq(u[N+1+i](zeta),i=1..N)];

(23)

> A:=genmatrix(eqns,Y,'b1');

array( 1 .. 20, 1 .. 20, [( 1, 8 ) = 0, ( 6, 12 ) = 0, ( 7, 18 ) = 0, ( 18, 7 ) = -1, ( 19, 4 ) = 0, ( 1, 4 ) = 0, ( 2, 4 ) = 0, ( 12, 6 ) = 0, ( 10, 14 ) = 0, ( 5, 20 ) = 0, ( 9, 14 ) = 0, ( 17, 9 ) ... (24)

> if N>2 then A:=map(evalf,A):end;

array( 1 .. 20, 1 .. 20, [( 1, 8 ) = 0., ( 6, 12 ) = 0., ( 7, 18 ) = 0., ( 18, 7 ) = -1., ( 19, 4 ) = 0., ( 1, 4 ) = 0., ( 2, 4 ) = 0., ( 12, 6 ) = 0., ( 10, 14 ) = 0., ( 5, 20 ) = 0., ( 9, 14 ) = 0.,... (25)

> evalm(A);

array( 1 .. 20, 1 .. 20, [( 1, 8 ) = 0., ( 6, 12 ) = 0., ( 7, 18 ) = 0., ( 18, 7 ) = -1., ( 19, 4 ) = 0., ( 1, 4 ) = 0., ( 2, 4 ) = 0., ( 12, 6 ) = 0., ( 10, 14 ) = 0., ( 5, 20 ) = 0., ( 9, 14 ) = 0.,... (26)

> b:=matrix(2*N,1):for i from 1 to 2*N do b[i,1]:=-b1[i];od:evalm(b);

(27)

> h:=eval(1/(N+1));

(28)

> J:=jordan(A,S);

(29)

> mat:=evalm(S&*exponential(J,zeta)&*inverse(S)):

> mat1:=evalm(subs(zeta=zeta-zeta1,evalm(mat))):

> b2:=evalm(subs(zeta=zeta1,evalm(b))):

> mat2:=evalm(mat1&*b2):

> mat2:=map(expand,mat2):

> mat3:=map(int,mat2,zeta1=0..zeta):

> Y0:=matrix(2*N,1);

(30)

> for i to N do Y0[i,1]:=p[i];od:

> for i to N do Y0[N+i,1]:=c[i]:od:

> evalm(Y0);

(31)

> Y:=evalm(mat&*Y0+mat3):

> sol0:=map(eval,evalm(subs(zeta=0,evalm(Y)))):

> sol1:=map(eval,evalm(subs(zeta=epsilon/h,evalm(Y)))):

> for i to N do Eq[i]:=subs(diff(u(x,y),y)=epsilon/h*c[i],u(x,y)=p[i],x=i*h,bc3);od;

(32)

> for i to N do Eq[N+i]:=evalf(subs(diff(u(x,y),y)=epsilon/h*sol1[N+i,1],u(x,y)=sol1[i,1],bc4));od;

(33)

> csol:=solve({seq(Eq[i],i=1..2*N)},{seq(c[i],i=1..N),seq(p[i],i=1..N)});

(34)

> assign(csol);

> Y:=map(eval,Y):

> for i from 1 to N do u[i](zeta):=eval((Y[i,1]));od:

> for i from 0 to N+1 do u[i](zeta):=eval(u[i](zeta));od:

> for i from 0 to N+1 do u[i](y):=eval(subs(zeta=epsilon*y/h,u[i](zeta)));od;

(35)

> for i from 0 to N+1 do
pl[i]:=line([0.3,0.98-abs(i-5.25)*0.14],[0.6,evalf(subs(y=0.6,u[i](y)))],thickness=1,linestyle=dot);
pt[i]:=textplot([0.3,0.98-abs(i-5.25)*0.14,typeset(u[i],"(y)")],align=left):
end do:

> pp:=plot([seq(u[i](y),i=0..N+1)],y=0..1);

(36)

> display([pp,seq(pl[i],i=0..N+1),seq(pt[i],i=0..N+1)],axes=boxed,thickness=3,title="Figure Exp. 6.1.",labels=[y,"u"]);

Plot_2d

> M:=10;

(37)

> T1:=[seq(evalf(i/M),i=0..M)];

(38)

> for j from 1 to M do P[j]:=plot([seq([h*i,evalf(subs(y=T1[j],evalf(u[i](y))))],i=0..N+1)],style=line,thickness=3,axes=boxed,view=[0..1,0..1.1]):od:

> P[M+1]:=plot([seq([h*i,evalf(subs(x=i*h,1))],i=0..N+1)],style=line,thickness=3,title="Figure Exp. 6.2.",axes=boxed):

> for j from 1 to M+1 do
pt[j]:=textplot([0.5,evalf(subs(y=T1[j],u[5](y))),typeset(y,sprintf("=%4.2f",T1[j]))],align=above);
od:

> display({seq(P[i],i=1..M+1),seq(pt[j],j=1..M+1)},labels=[x,u]);

Plot_2d

> Ny:=30;

(39)

> PP:=matrix(N+2,Ny);

(40)

> for i to Ny do PP[1,i]:=0;PP[N+2,i]:=0;od:

> for i to N+2 do PP[i,1]:=0;PP[i,Ny]:=1;od:

> for i from 2 to N+1 do for j from 2 to Ny-1 do PP[i,j]:=evalf(subs(y=(j-1)/(Ny-1),u[i-1](y)));od;od:

> plotdata := [seq([ seq([(i-1)/(N+1),(j-1)/(Ny-1),PP[i,j]], i=1..N+2)], j=1..Ny)]:

> surfdata(plotdata,axes=boxed,title="Figure Exp. 6.3.",labels=[x,y,u],orientation=[-120,60] );

Plot