Example 6.3
| > | restart;with(linalg):with(plots): |
The governing equation is entered in the following form:
| > | ge:=diff(u(x,y),y$2)=2*Pe*(1-x^2)*diff(u(x,y),y)-diff(u(x,y),x$2)-1/x*diff(u(x,y),x); |
| (1) |
| > | Digits:=30; |
| (2) |
For this exmaple, Digits has to be set to 30 for accurate predicitons. The boundary conditions are enterered as:
| > | bc1:=diff(u(x,y),x); |
| (3) |
| > | bc2:=u(x,y)-1; |
| (4) |
| > | bc3:=u(x,y)-0; |
| (5) |
| > | bc4:=diff(u(x,y),y); |
| (6) |
Parameters are entered here:
| > | Pe:=1.0; |
| (7) |
| > | epsilon:=1; |
| (8) |
| > | L:=1; |
| (9) |
| > | H:=2; |
| (10) |
| > | dydxf:=1/2/h*(-u[m+2](zeta)-3*u[m](zeta)+4*u[m+1](zeta)): |
| > | dydxb:=1/2/h*(u[m-2](zeta)+3*u[m](zeta)-4*u[m-1](zeta)): |
| > | dydx:=1/2/h*(u[m+1](zeta)-u[m-1](zeta)): |
| > | d2ydx2:=1/h^2*(u[m-1](zeta)-2*u[m](zeta)+u[m+1](zeta)): |
| > | bc1:=subs(diff(u(x,y),x)=subs(m=0,dydxf),u(x,y)=u[0](zeta),x=0,bc1): |
| > | bc2:=subs(diff(u(x,y),x)=subs(m=N+1,dydxb),u(x,y)=u[N+1](zeta),x=1,bc2): |
| > | N:=2; |
| (11) |
| > | eq[0]:=bc1; |
| (12) |
| > | eq[N+1]:=bc2; |
| (13) |
| > | 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),diff(u(x,y),y)=epsilon/h*u[N+1+i](zeta),u(x,y)=u[i](zeta),x=i*h,-rhs(h^2/epsilon^2*ge));od; |
| (14) |
| > | u[0](zeta):=(solve(eq[0],u[0](zeta))); |
| (15) |
| > | u[N+1](zeta):=solve(eq[N+1],u[N+1](zeta)); |
| (16) |
| > | for i from 1 to N do eq[i]:=diff(u[i](zeta),zeta)= -u[N+1+i](zeta);od; |
| (17) |
| > | h:=L/(N+1); |
| (18) |
| > | 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; |
| (19) |
| > | eqns:=[seq(rhs(eq[j]),j=1..N),seq(rhs(eq[N+1+j]),j=1..N)]: |
| > | Y:=[seq(u[i](zeta),i=1..N),seq(u[N+1+i](zeta),i=1..N)]; |
| (20) |
| > | A:=genmatrix(eqns,Y,'b1'): |
| > | Nrow:=rowdim(A); |
| (21) |
| > | II:=eigenvalues(A): |
| > | for i to Nrow do
lambda[i]:=II[i]; end do: |
| > | Id:=Matrix(Nrow,Nrow,shape=identity): |
| > | X:=matrix(Nrow,1,[seq(beta[i],i=1..Nrow)]): |
| > | for k to Nrow do
G:=evalm((A-lambda[k]*Id)&*X); eqx[1]:=beta[1]=1: for i from 2 to Nrow do eqx[i]:=G[i-1,1]: end do: cons:=fsolve({seq(eqx[i],i=1..Nrow)},{seq(beta[i],i=1..Nrow)}): assign(cons): XX[k]:=map(eval,evalm(X)): for i to Nrow do unassign('beta[i]'): end do: end do: |
| > | PV:=Matrix(Nrow,Nrow,[seq(evalm(XX[i]),i=1..Nrow)]): |
| > | expD1:=Matrix(1..Nrow,1..Nrow,shape=diagonal): |
| > | for i to Nrow do
expD1[i,i]:=exp(lambda[i]*zeta): end do: |
| > | mat:=evalm(PV&*expD1&*inverse(PV)): |
| > |
| > | if N>2 then A:=map(evalf,A):end: |
| > | b:=matrix(2*N,1):for i from 1 to 2*N do b[i,1]:=-b1[i];od:evalm(b): |
| > | h:=eval(1/(N+1)); |
| (22) |
| > |
| > | 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); |
| (23) |
| > | 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); |
| (24) |
| > | Y:=evalm(mat&*Y0+mat3): |
| > | sol0:=map(eval,evalm(subs(zeta=0,evalm(Y)))): |
| > | sol1:=map(eval,evalm(subs(zeta=epsilon*H/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; |
| (25) |
| > | 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: |
| > | csol:=solve({seq(Eq[i],i=1..2*N)},{seq(c[i],i=1..N),seq(p[i],i=1..N)}); |
| (26) |
| > | 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: |
| > | for i from 0 to N+1 by 2 do p[i]:=plot(u[i](y),y=0..H,thickness=3);od: |
| > | pp:=plot([seq(u[i](y),i=0..N+1,2)],y=0..H,thickness=3,legend=[seq(typeset(u[i],"(y)"),i=0..N+1,2)]); |
| (27) |
| > | display(pp,axes=boxed,title="Figure Exp.6.7.",labels=[y,"u"]); |
![]() |
| > | M:=5; |
| (28) |
| > | T1:=[seq(evalf(i*H/M),i=0..M)]: |
| > | P[1]:=plot([seq([h*i,0.],i=0..N+1)],style=line,thickness=3,axes=boxed): |
| > | for j from 2 to M+1 do P[j]:=plot([seq([h*i,evalf(subs(y=T1[j],evalf(u[i](y))))],i=0..N+1)],style=line,thickness=3,title="Figure Exp. 6.8.",axes=boxed):od: |
| > | 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,left}); od: |
| > | display({seq(P[i],i=1..M+1),seq(pt[i],i=1..M+1)},labels=[x,u]); |
| Error:TEXT location must be numeric; received: [.5, u[5](1.60000000000000000000000000000)] |
| > | a:=convert(T1[2],string); |
| (29) |
| > | a1:=sprintf("%4.2f",T1[2]); |
| (30) |
| > | Ny:=30; |
| (31) |
| > | PP:=matrix(N+2,Ny); |
| (32) |
| > | for i from 1 to N+2 do for j from 1 to Ny do PP[i,j]:=evalf(subs(y=(j-1)*H/(Ny-1),u[i-1](y)));od;od: |
| > | PP[N+2,1]:=0; |
| (33) |
| > | plotdata := [seq([ seq([(i-1)/(N+1),(j-1)*H/(Ny-1),PP[i,j]], i=1..N+2)], j=1..Ny)]: |
| > | surfdata(plotdata,axes=boxed,title="Figure Exp. 6.9.", labels=[x,y,u],orientation=[-150,45] ); |
![]() |
| > |