Example 6.1 Rev 1N2.mw

Example 6.1 N2

> |

Enter the governing equation:

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

| (1)

Enter the boundary conditions:

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

| (2)

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

| (3)

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

| (4)

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

| (5)

> | epsilon:=1;

| (6)

Enter the finite difference approximations for the derivatives:

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

| (7)

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

| (8)

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

| (9)

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

| (10)

Convert the boundary conditions to finite difference form:

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

| (11)

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

| (12)

Enter the number of interior node points:

> | N:=2;

| (13)

> | eq[0]:=bc1;

| (14)

> | eq[N+1]:=bc2;

| (15)

Convert the governing equation to finite difference from (equation 6.1.12):

> | 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;

| (16)

Enter the boundary values:

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

| (17)

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

| (18)

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

| (19)

> | 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;

| (20)

Generate the A matrix using the governing equations and the dependent variables

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

| (21)

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

| (22)

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

| (23)

Convert the entries of the A matrix as decimals if N is greater than two (as in chapter 5.1):

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

> |

| (24)

> |

Error, illegal use of an object as a name |

> | evalm(A);

| (25)

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

| (26)

Note that for the example given the b vector is zero. However, depending on the boundary conditions, bc3 and bc3, the b vector can be a function of ζ or a constant vector.

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

| (27)

> | J:=jordan(A,S);

| (28)

> | 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):

The initial condition vector is defined here.

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

| (29)

> | 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);

| (30)

The solution is found by adding the nonhomogeneous part to the homogeneous part.

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

The solution at y = 0 and y = 1 is stored in sol0 and sol1 to calculate the unknown constants.

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

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

Now the boundary conditions bc3 and bc4 are applied.

> | 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;

| (31)

> | 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;

| (32)

The unknown constants are solved as:

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

| (33)

> | 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;

| (34)

Hence, the semianalytical solution is obtained for temperature distribution. The plots obtained for N=10 node points is given below:

> | 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);

| (35)

> | 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"]);

> | M:=10;

| (36)

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

| (37)

> | 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]);

Error:TEXT location must be numeric; received: [.5, u[5](.1000000000)] |

> | Ny:=30;

| (38)

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

| (39)

> | 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] );

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