Example 6.2 Rev 1b.mw

Example 6.2 b 

> restart;with(plottools):with(linalg):with(plots):
 

> Digits:=12;
 

12 (1)
 

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

diff(diff(u(x, y), y), y) = `+`(`-`(`*`(`^`(epsilon, 2), `*`(diff(diff(u(x, y), x), x))))) (2)
 

> bc1:=u(x,y)-0;
 

u(x, y) (3)
 

> bc2:=u(x,y)-0;
 

u(x, y) (4)
 

Now, the boundary condition at y = 0 is redefined. 

> bc3:=diff(u(x,y),y)-u(x,y);
 

`+`(diff(u(x, y), y), `-`(u(x, y))) (5)
 

> bc4:=u(x,y)-1;
 

`+`(u(x, y), `-`(1)) (6)
 

> epsilon:=1;
 

1 (7)
 

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

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(u[`+`(m, 2)](zeta)), `-`(`*`(3, `*`(u[m](zeta)))), `*`(4, `*`(u[`+`(m, 1)](zeta)))))), `*`(h))) (8)
 

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

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(u[`+`(m, `-`(2))](zeta), `*`(3, `*`(u[m](zeta))), `-`(`*`(4, `*`(u[`+`(m, `-`(1))](zeta))))))), `*`(h))) (9)
 

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

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(u[`+`(m, 1)](zeta), `-`(u[`+`(m, `-`(1))](zeta))))), `*`(h))) (10)
 

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

`/`(`*`(`+`(u[`+`(m, `-`(1))](zeta), `-`(`*`(2, `*`(u[m](zeta)))), u[`+`(m, 1)](zeta))), `*`(`^`(h, 2))) (11)
 

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

u[0](zeta) (12)
 

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

u[`+`(N, 1)](zeta) (13)
 

> N:=10;
 

10 (14)
 

> eq[0]:=bc1;
 

u[0](zeta) (15)
 

> eq[N+1]:=bc2;
 

u[11](zeta) (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;
 

 

 

 

 

 

 

 

 

 

diff(u[12](zeta), zeta) = `+`(`-`(u[0](zeta)), `*`(2, `*`(u[1](zeta))), `-`(u[2](zeta)))
diff(u[13](zeta), zeta) = `+`(`-`(u[1](zeta)), `*`(2, `*`(u[2](zeta))), `-`(u[3](zeta)))
diff(u[14](zeta), zeta) = `+`(`-`(u[2](zeta)), `*`(2, `*`(u[3](zeta))), `-`(u[4](zeta)))
diff(u[15](zeta), zeta) = `+`(`-`(u[3](zeta)), `*`(2, `*`(u[4](zeta))), `-`(u[5](zeta)))
diff(u[16](zeta), zeta) = `+`(`-`(u[4](zeta)), `*`(2, `*`(u[5](zeta))), `-`(u[6](zeta)))
diff(u[17](zeta), zeta) = `+`(`-`(u[5](zeta)), `*`(2, `*`(u[6](zeta))), `-`(u[7](zeta)))
diff(u[18](zeta), zeta) = `+`(`-`(u[6](zeta)), `*`(2, `*`(u[7](zeta))), `-`(u[8](zeta)))
diff(u[19](zeta), zeta) = `+`(`-`(u[7](zeta)), `*`(2, `*`(u[8](zeta))), `-`(u[9](zeta)))
diff(u[20](zeta), zeta) = `+`(`-`(u[8](zeta)), `*`(2, `*`(u[9](zeta))), `-`(u[10](zeta)))
diff(u[21](zeta), zeta) = `+`(`-`(u[9](zeta)), `*`(2, `*`(u[10](zeta))), `-`(u[11](zeta))) (17)
 

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

0 (18)
 

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

0 (19)
 

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

 

 

 

 

 

 

 

 

 

diff(u[1](zeta), zeta) = u[12](zeta)
diff(u[2](zeta), zeta) = u[13](zeta)
diff(u[3](zeta), zeta) = u[14](zeta)
diff(u[4](zeta), zeta) = u[15](zeta)
diff(u[5](zeta), zeta) = u[16](zeta)
diff(u[6](zeta), zeta) = u[17](zeta)
diff(u[7](zeta), zeta) = u[18](zeta)
diff(u[8](zeta), zeta) = u[19](zeta)
diff(u[9](zeta), zeta) = u[20](zeta)
diff(u[10](zeta), zeta) = u[21](zeta) (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;
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

diff(u[1](zeta), zeta) = u[12](zeta)
diff(u[2](zeta), zeta) = u[13](zeta)
diff(u[3](zeta), zeta) = u[14](zeta)
diff(u[4](zeta), zeta) = u[15](zeta)
diff(u[5](zeta), zeta) = u[16](zeta)
diff(u[6](zeta), zeta) = u[17](zeta)
diff(u[7](zeta), zeta) = u[18](zeta)
diff(u[8](zeta), zeta) = u[19](zeta)
diff(u[9](zeta), zeta) = u[20](zeta)
diff(u[10](zeta), zeta) = u[21](zeta)
diff(u[12](zeta), zeta) = `+`(`*`(2, `*`(u[1](zeta))), `-`(u[2](zeta)))
diff(u[13](zeta), zeta) = `+`(`-`(u[1](zeta)), `*`(2, `*`(u[2](zeta))), `-`(u[3](zeta)))
diff(u[14](zeta), zeta) = `+`(`-`(u[2](zeta)), `*`(2, `*`(u[3](zeta))), `-`(u[4](zeta)))
diff(u[15](zeta), zeta) = `+`(`-`(u[3](zeta)), `*`(2, `*`(u[4](zeta))), `-`(u[5](zeta)))
diff(u[16](zeta), zeta) = `+`(`-`(u[4](zeta)), `*`(2, `*`(u[5](zeta))), `-`(u[6](zeta)))
diff(u[17](zeta), zeta) = `+`(`-`(u[5](zeta)), `*`(2, `*`(u[6](zeta))), `-`(u[7](zeta)))
diff(u[18](zeta), zeta) = `+`(`-`(u[6](zeta)), `*`(2, `*`(u[7](zeta))), `-`(u[8](zeta)))
diff(u[19](zeta), zeta) = `+`(`-`(u[7](zeta)), `*`(2, `*`(u[8](zeta))), `-`(u[9](zeta)))
diff(u[20](zeta), zeta) = `+`(`-`(u[8](zeta)), `*`(2, `*`(u[9](zeta))), `-`(u[10](zeta)))
diff(u[21](zeta), zeta) = `+`(`-`(u[9](zeta)), `*`(2, `*`(u[10](zeta)))) (21)
 

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

[u[12](zeta), u[13](zeta), u[14](zeta), u[15](zeta), u[16](zeta), u[17](zeta), u[18](zeta), u[19](zeta), u[20](zeta), u[21](zeta), `+`(`*`(2, `*`(u[1](zeta))), `-`(u[2](zeta))), `+`(`-`(u[1](zeta)), `...
[u[12](zeta), u[13](zeta), u[14](zeta), u[15](zeta), u[16](zeta), u[17](zeta), u[18](zeta), u[19](zeta), u[20](zeta), u[21](zeta), `+`(`*`(2, `*`(u[1](zeta))), `-`(u[2](zeta))), `+`(`-`(u[1](zeta)), `...
[u[12](zeta), u[13](zeta), u[14](zeta), u[15](zeta), u[16](zeta), u[17](zeta), u[18](zeta), u[19](zeta), u[20](zeta), u[21](zeta), `+`(`*`(2, `*`(u[1](zeta))), `-`(u[2](zeta))), `+`(`-`(u[1](zeta)), `...
[u[12](zeta), u[13](zeta), u[14](zeta), u[15](zeta), u[16](zeta), u[17](zeta), u[18](zeta), u[19](zeta), u[20](zeta), u[21](zeta), `+`(`*`(2, `*`(u[1](zeta))), `-`(u[2](zeta))), `+`(`-`(u[1](zeta)), `...
(22)
 

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

[u[1](zeta), u[2](zeta), u[3](zeta), u[4](zeta), u[5](zeta), u[6](zeta), u[7](zeta), u[8](zeta), u[9](zeta), u[10](zeta), u[12](zeta), u[13](zeta), u[14](zeta), u[15](zeta), u[16](zeta), u[17](zeta), ...
[u[1](zeta), u[2](zeta), u[3](zeta), u[4](zeta), u[5](zeta), u[6](zeta), u[7](zeta), u[8](zeta), u[9](zeta), u[10](zeta), u[12](zeta), u[13](zeta), u[14](zeta), u[15](zeta), u[16](zeta), u[17](zeta), ...
(23)
 

> A:=genmatrix(eqns,Y,'b1');
 

array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 14, 13 ) = 0, ( 19, 10 ) = -1, ( 4, 3 ) = 0, ( 20, 10 ) = 2, ( 5, 12 ) = 0, ( 16... (24)
 

> if N>2 then A:=map(evalf,A):end;
 

array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0., ( 12, 20 ) = 0., ( 13, 18 ) = 0., ( 18, 16 ) = 0., ( 11, 9 ) = 0., ( 2, 14 ) = 0., ( 14, 13 ) = 0., ( 19, 10 ) = -1., ( 4, 3 ) = 0., ( 20, 10 ) = 2., ( 5, 12 )... (25)
 

> evalm(A);
 

array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0., ( 12, 20 ) = 0., ( 13, 18 ) = 0., ( 18, 16 ) = 0., ( 11, 9 ) = 0., ( 2, 14 ) = 0., ( 14, 13 ) = 0., ( 19, 10 ) = -1., ( 4, 3 ) = 0., ( 20, 10 ) = 2., ( 5, 12 )... (26)
 

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

array( 1 .. 20, 1 .. 1, [( 5, 1 ) = 0, ( 12, 1 ) = 0, ( 17, 1 ) = 0, ( 8, 1 ) = 0, ( 19, 1 ) = 0, ( 13, 1 ) = 0, ( 4, 1 ) = 0, ( 2, 1 ) = 0, ( 7, 1 ) = 0, ( 6, 1 ) = 0, ( 11, 1 ) = 0, ( 9, 1 ) = 0, ( ... (27)
 

> h:=eval(1/(N+1));
 

`/`(1, 11) (28)
 

> J:=jordan(A,S);
 

array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
array( 1 .. 20, 1 .. 20, [( 5, 1 ) = 0, ( 12, 20 ) = 0, ( 13, 18 ) = 0, ( 18, 16 ) = 0, ( 11, 9 ) = 0, ( 2, 14 ) = 0, ( 19, 10 ) = 0, ( 14, 13 ) = 0, ( 4, 3 ) = 0, ( 20, 10 ) = 0, ( 5, 12 ) = 0, ( 16,...
(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);
 

array( 1 .. 20, 1 .. 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);
 

array( 1 .. 20, 1 .. 1, [( 5, 1 ) = p[5], ( 12, 1 ) = c[2], ( 17, 1 ) = c[7], ( 8, 1 ) = p[8], ( 19, 1 ) = c[9], ( 13, 1 ) = c[3], ( 4, 1 ) = p[4], ( 2, 1 ) = p[2], ( 7, 1 ) = p[7], ( 6, 1 ) = p[6], (... (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)))):
 

>
 

> bc3:=diff(u(x,y),y)-u(x,y);
 

`+`(diff(u(x, y), y), `-`(u(x, y))) (32)
 

> 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;
 

 

 

 

 

 

 

 

 

 

`+`(`*`(11, `*`(c[1])), `-`(p[1]))
`+`(`*`(11, `*`(c[2])), `-`(p[2]))
`+`(`*`(11, `*`(c[3])), `-`(p[3]))
`+`(`*`(11, `*`(c[4])), `-`(p[4]))
`+`(`*`(11, `*`(c[5])), `-`(p[5]))
`+`(`*`(11, `*`(c[6])), `-`(p[6]))
`+`(`*`(11, `*`(c[7])), `-`(p[7]))
`+`(`*`(11, `*`(c[8])), `-`(p[8]))
`+`(`*`(11, `*`(c[9])), `-`(p[9]))
`+`(`*`(11, `*`(c[10])), `-`(p[10])) (33)
 

> 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:
 

The new sets of constants are: 

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

{p[6] = 0.827183801643e-1, p[4] = 0.760595917920e-1, p[9] = 0.452952522478e-1, p[1] = 0.236225323780e-1, p[8] = 0.632520997147e-1, p[5] = 0.827183825965e-1, p[2] = 0.452954269030e-1, c[9] = 0.41177502...
{p[6] = 0.827183801643e-1, p[4] = 0.760595917920e-1, p[9] = 0.452952522478e-1, p[1] = 0.236225323780e-1, p[8] = 0.632520997147e-1, p[5] = 0.827183825965e-1, p[2] = 0.452954269030e-1, c[9] = 0.41177502...
{p[6] = 0.827183801643e-1, p[4] = 0.760595917920e-1, p[9] = 0.452952522478e-1, p[1] = 0.236225323780e-1, p[8] = 0.632520997147e-1, p[5] = 0.827183825965e-1, p[2] = 0.452954269030e-1, c[9] = 0.41177502...
{p[6] = 0.827183801643e-1, p[4] = 0.760595917920e-1, p[9] = 0.452952522478e-1, p[1] = 0.236225323780e-1, p[8] = 0.632520997147e-1, p[5] = 0.827183825965e-1, p[2] = 0.452954269030e-1, c[9] = 0.41177502...
{p[6] = 0.827183801643e-1, p[4] = 0.760595917920e-1, p[9] = 0.452952522478e-1, p[1] = 0.236225323780e-1, p[8] = 0.632520997147e-1, p[5] = 0.827183825965e-1, p[2] = 0.452954269030e-1, c[9] = 0.41177502...
(34)
 

> assign(csol);
 

Using the new values of constants the semianalytical solution is recalculated and following plots are obtained. 

> YY:=map(eval,Y):
 

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

 

 

 

 

 

 

 

 

 

 

 

0
`+`(`*`(0.19593e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `*`(0.21334560e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y))))))), `*`(0.155451316190e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))))), ...
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`+`(`-`(`*`(0.32982e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y))))))), `-`(`*`(0.36207855e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.298308467346e-1, `*`(exp(`+`(`*`(3.13092644202, `*`...
`+`(`-`(`*`(0.32982e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y))))))), `-`(`*`(0.36207855e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.298308467346e-1, `*`(exp(`+`(`*`(3.13092644202, `*`...
`+`(`-`(`*`(0.32982e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y))))))), `-`(`*`(0.36207855e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.298308467346e-1, `*`(exp(`+`(`*`(3.13092644202, `*`...
`+`(`*`(0.19622e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `-`(`*`(0.21508854e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.155451185550e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))...
`+`(`*`(0.19622e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `-`(`*`(0.21508854e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.155451185550e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))...
`+`(`*`(0.19622e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `-`(`*`(0.21508854e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.155451185550e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))...
`+`(`*`(0.19622e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `-`(`*`(0.21508854e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.155451185550e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))...
`+`(`*`(0.19622e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `-`(`*`(0.21508854e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.155451185550e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))...
`+`(`*`(0.19622e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `-`(`*`(0.21508854e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.155451185550e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))...
`+`(`*`(0.19622e-10, `*`(exp(`+`(`*`(21.1088456996, `*`(y)))))), `-`(`*`(0.21508854e-7, `*`(exp(`+`(`-`(`*`(6.19811625046, `*`(y)))))))), `*`(0.155451185550e-1, `*`(exp(`+`(`*`(3.13092644202, `*`(y)))...
0 (35)
 

The constants are unassigned. 

> for i to N do unassign('c[i]'):unassign('p[i]'):od:
 

> 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:
 

> for i from 0 to N+1 do p[i]:=plot(u[i](y),y=0..1,thickness=3);od:
 

> pp:=plot([seq(u[i](y),i=0..N+1)],y=0..1,thickness=3):
 

> display([pp,seq(pl[i],i=0..N+1),seq(pt[i],i=0..N+1)],title="Figure Exp. 6.4.",axes=boxed,labels=[y,"u"]);
 

Plot_2d
 

> M:=5;
 

5 (36)
 

> T1:=[seq(evalf(i/M),i=0..M)];
 

[0., .200000000000, .400000000000, .600000000000, .800000000000, 1.] (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. 6.5.",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[i],i=1..M+1)},labels=[x,u]);
 

Plot_2d
 

> Ny:=20;
 

20 (38)
 

> PP:=matrix(N+2,Ny);
 

array( 1 .. 12, 1 .. 20, [ ] ) (39)
 

For the three dimensional plot, first the boundaries, x = 0, x = 1, and y = 1 are defined. 

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

> for i to N+2 do PP[i,Ny]:=1;od:
 

The temperature inside the rectangle is obtained using the semianalytical solution. 

> for i from 2 to N+1 do for j from 1 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.6.6.", labels=[x,y,u],orientation=[-120,60] );
 

Plot
 

>
 

>