Example5.2.2 Rev 1.mw

Example 5.2.2 Variable Diffusivity 

> restart;
 

> with(linalg):with(plots):
 

> ge:=diff(u(x,t),t)=diff((1+alpha*u(x,t))*diff(u(x,t),x),x);
 

diff(u(x, t), t) = `+`(`*`(alpha, `*`(`^`(diff(u(x, t), x), 2))), `*`(`+`(1, `*`(alpha, `*`(u(x, t)))), `*`(diff(diff(u(x, t), x), x)))) (1)
 

> bc1:=u(x,t)-1;
 

`+`(u(x, t), `-`(1)) (2)
 

> bc2:=diff(u(x,t),x);
 

diff(u(x, t), x) (3)
 

> IC:=u(x,0)=0;
 

u(x, 0) = 0 (4)
 

> N:=10;
 

10 (5)
 

> L:=1;
 

1 (6)
 

> alpha:=1;
 

1 (7)
 

> dydxf:=1/2*(-u[2](t)-3*u[0](t)+4*u[1](t))/h;
 

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(u[2](t)), `-`(`*`(3, `*`(u[0](t)))), `*`(4, `*`(u[1](t)))))), `*`(h))) (8)
 

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

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(u[9](t), `*`(3, `*`(u[11](t))), `-`(`*`(4, `*`(u[10](t))))))), `*`(h))) (9)
 

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

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

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

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

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

`+`(u[0](t), `-`(1)) (12)
 

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

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(u[9](t), `*`(3, `*`(u[11](t))), `-`(`*`(4, `*`(u[10](t))))))), `*`(h))) (13)
 

> eq[0]:=bc1;
 

`+`(u[0](t), `-`(1)) (14)
 

> eq[N+1]:=bc2;
 

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(u[9](t), `*`(3, `*`(u[11](t))), `-`(`*`(4, `*`(u[10](t))))))), `*`(h))) (15)
 

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

 

 

 

 

 

 

 

 

 

diff(u[1](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[2](t), `-`(u[0](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[1](t)), `*`(`+`(u[0](t), `-`(`*`(2, `*`(u[1](t)))), u[2](t)))), `*`(`^`(h, 2))))
diff(u[2](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[3](t), `-`(u[1](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[2](t)), `*`(`+`(u[1](t), `-`(`*`(2, `*`(u[2](t)))), u[3](t)))), `*`(`^`(h, 2))))
diff(u[3](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[4](t), `-`(u[2](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[3](t)), `*`(`+`(u[2](t), `-`(`*`(2, `*`(u[3](t)))), u[4](t)))), `*`(`^`(h, 2))))
diff(u[4](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[5](t), `-`(u[3](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[4](t)), `*`(`+`(u[3](t), `-`(`*`(2, `*`(u[4](t)))), u[5](t)))), `*`(`^`(h, 2))))
diff(u[5](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[6](t), `-`(u[4](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[5](t)), `*`(`+`(u[4](t), `-`(`*`(2, `*`(u[5](t)))), u[6](t)))), `*`(`^`(h, 2))))
diff(u[6](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[7](t), `-`(u[5](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[6](t)), `*`(`+`(u[5](t), `-`(`*`(2, `*`(u[6](t)))), u[7](t)))), `*`(`^`(h, 2))))
diff(u[7](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[8](t), `-`(u[6](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[7](t)), `*`(`+`(u[6](t), `-`(`*`(2, `*`(u[7](t)))), u[8](t)))), `*`(`^`(h, 2))))
diff(u[8](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[9](t), `-`(u[7](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[8](t)), `*`(`+`(u[7](t), `-`(`*`(2, `*`(u[8](t)))), u[9](t)))), `*`(`^`(h, 2))))
diff(u[9](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[10](t), `-`(u[8](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[9](t)), `*`(`+`(u[8](t), `-`(`*`(2, `*`(u[9](t)))), u[10](t)))), `*`(`^`(h, 2))))
diff(u[10](t), t) = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(u[11](t), `-`(u[9](t))), 2))), `*`(`^`(h, 2))), `/`(`*`(`+`(1, u[10](t)), `*`(`+`(u[9](t), `-`(`*`(2, `*`(u[10](t)))), u[11](t)))), `*`(`^`(h, 2)... (16)
 

> u[0](t):=(solve(eq[0],u[0](t)));
 

1 (17)
 

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

`+`(`-`(`*`(`/`(1, 3), `*`(u[9](t)))), `*`(`/`(4, 3), `*`(u[10](t)))) (18)
 

> h:=L/(N+1);
 

`/`(1, 11) (19)
 

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

 

 

 

 

 

 

 

 

 

diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t))))))
diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[3](t), `-`(u[1](t))), 2))), `*`(121, `*`(`+`(1, u[2](t)), `*`(`+`(u[1](t), `-`(`*`(2, `*`(u[2](t)))), u[3](t))))))
diff(u[3](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[4](t), `-`(u[2](t))), 2))), `*`(121, `*`(`+`(1, u[3](t)), `*`(`+`(u[2](t), `-`(`*`(2, `*`(u[3](t)))), u[4](t))))))
diff(u[4](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[5](t), `-`(u[3](t))), 2))), `*`(121, `*`(`+`(1, u[4](t)), `*`(`+`(u[3](t), `-`(`*`(2, `*`(u[4](t)))), u[5](t))))))
diff(u[5](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[6](t), `-`(u[4](t))), 2))), `*`(121, `*`(`+`(1, u[5](t)), `*`(`+`(u[4](t), `-`(`*`(2, `*`(u[5](t)))), u[6](t))))))
diff(u[6](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[7](t), `-`(u[5](t))), 2))), `*`(121, `*`(`+`(1, u[6](t)), `*`(`+`(u[5](t), `-`(`*`(2, `*`(u[6](t)))), u[7](t))))))
diff(u[7](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[8](t), `-`(u[6](t))), 2))), `*`(121, `*`(`+`(1, u[7](t)), `*`(`+`(u[6](t), `-`(`*`(2, `*`(u[7](t)))), u[8](t))))))
diff(u[8](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[9](t), `-`(u[7](t))), 2))), `*`(121, `*`(`+`(1, u[8](t)), `*`(`+`(u[7](t), `-`(`*`(2, `*`(u[8](t)))), u[9](t))))))
diff(u[9](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[10](t), `-`(u[8](t))), 2))), `*`(121, `*`(`+`(1, u[9](t)), `*`(`+`(u[8](t), `-`(`*`(2, `*`(u[9](t)))), u[10](t))))))
diff(u[10](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(`-`(`*`(`/`(4, 3), `*`(u[9](t)))), `*`(`/`(4, 3), `*`(u[10](t)))), 2))), `*`(121, `*`(`+`(1, u[10](t)), `*`(`+`(`*`(`/`(2, 3), `*`(u[9](t))), `-`(`... (20)
 

> eqs:=seq((eq[j]),j=1..N);
 

diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
diff(u[1](t), t) = `+`(`*`(`/`(121, 4), `*`(`^`(`+`(u[2](t), `-`(1)), 2))), `*`(121, `*`(`+`(1, u[1](t)), `*`(`+`(1, `-`(`*`(2, `*`(u[1](t)))), u[2](t)))))), diff(u[2](t), t) = `+`(`*`(`/`(121, 4), `*...
(21)
 

> Y:=seq(u[i](t),i=1..N);
 

u[1](t), u[2](t), u[3](t), u[4](t), u[5](t), u[6](t), u[7](t), u[8](t), u[9](t), u[10](t) (22)
 

> ICs:=seq(u[i](0)=rhs(IC),i=1..N);
 

u[1](0) = 0, u[2](0) = 0, u[3](0) = 0, u[4](0) = 0, u[5](0) = 0, u[6](0) = 0, u[7](0) = 0, u[8](0) = 0, u[9](0) = 0, u[10](0) = 0 (23)
 

> sol:=dsolve({eqs,ICs},{Y},type=numeric,output=listprocedure);
 

[t = proc (t) local res, data, solnproc, outpoint, t; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; `:=`(_EnvDSNumericSaveDigits, Digits); `:=`(Digits, 14); if _EnvInFsolve ...
[t = proc (t) local res, data, solnproc, outpoint, t; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; `:=`(_EnvDSNumericSaveDigits, Digits); `:=`(Digits, 14); if _EnvInFsolve ...
[t = proc (t) local res, data, solnproc, outpoint, t; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; `:=`(_EnvDSNumericSaveDigits, Digits); `:=`(Digits, 14); if _EnvInFsolve ...
[t = proc (t) local res, data, solnproc, outpoint, t; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; `:=`(_EnvDSNumericSaveDigits, Digits); `:=`(Digits, 14); if _EnvInFsolve ...
[t = proc (t) local res, data, solnproc, outpoint, t; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; `:=`(_EnvDSNumericSaveDigits, Digits); `:=`(Digits, 14); if _EnvInFsolve ...
(24)
 

> for i to N do U[i]:=subs(sol,u[i](t));od:
 

> U[0]:=subs(u[1](t)=U[1],u[2](t)=U[2],u[0](t));
 

1 (25)
 

> U[N+1]:=subs(u[N](t)=U[N],u[N-1](t)=U[N-1],u[N+1](t));
 

`+`(`-`(`*`(`/`(1, 3), `*`(U[9]))), `*`(`/`(4, 3), `*`(U[10]))) (26)
 

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

> pp:=plot([seq(U[i](t),i=0..N+1)],t=0..0.4);
 

PLOT(CURVES([[0., 1.], [0.871886166666666594e-2, 1.], [0.163050965833333346e-1, 1.], [0.248366111666666682e-1, 1.], [0.334246781666666660e-1, 1.], [0.419719275833333322e-1, 1.], [0.498963219166666666e... (27)
 

> display(pp,axes=boxed,title="Figure Exp. 5.2.3.",thickness=3,labels=[t,"u"]);
 

Plot_2d
 

> tf:=1.;
 

1. (28)
 

> M:=30;
 

30 (29)
 

> T1:=[seq(tf*i/M,i=0..M)];
 

[0., 0.3333333333e-1, 0.6666666667e-1, .1000000000, .1333333333, .1666666667, .2000000000, .2333333333, .2666666667, .3000000000, .3333333333, .3666666667, .4000000000, .4333333333, .4666666667, .5000...
[0., 0.3333333333e-1, 0.6666666667e-1, .1000000000, .1333333333, .1666666667, .2000000000, .2333333333, .2666666667, .3000000000, .3333333333, .3666666667, .4000000000, .4333333333, .4666666667, .5000...
[0., 0.3333333333e-1, 0.6666666667e-1, .1000000000, .1333333333, .1666666667, .2000000000, .2333333333, .2666666667, .3000000000, .3333333333, .3666666667, .4000000000, .4333333333, .4666666667, .5000...
[0., 0.3333333333e-1, 0.6666666667e-1, .1000000000, .1333333333, .1666666667, .2000000000, .2333333333, .2666666667, .3000000000, .3333333333, .3666666667, .4000000000, .4333333333, .4666666667, .5000...
[0., 0.3333333333e-1, 0.6666666667e-1, .1000000000, .1333333333, .1666666667, .2000000000, .2333333333, .2666666667, .3000000000, .3333333333, .3666666667, .4000000000, .4333333333, .4666666667, .5000...
(30)
 

> PP:=matrix(N+2,M+1);
 

array( 1 .. 12, 1 .. 31, [ ] ) (31)
 

> for i from 1 to N+2 do PP[i,1]:=evalf(subs(x=(i-1)*h,rhs(IC)));od:
 

> for i from 1 to N+2 do for j from 2 to M+1 do PP[i,j]:=evalf(subs(t=T1[j],U[i-1](t)));od;od:
 

> plotdata := [seq([ seq([(i-1)*h,T1[j],PP[i,j]], i=1..N+2)], j=1..M+1)]:
 

> surfdata(plotdata,axes=boxed,title="Figure Exp. 5.2.4.",labels=[x,t,u],orientation=[-45,60]);
 

Plot
 

>