Example 5.11 

> restart;
 

> with(linalg):with(plots):
 

> ge:=diff(u(x,t),t)=-diff(u(x,t),x);
 

diff(u(x, t), t) = `+`(`-`(diff(u(x, t), x))) (1)
 

> bc1:=u(x,t)-1;
 

`+`(u(x, t), `-`(1)) (2)
 

> bc2:=diff(u(x,t),t)=-diff(u(x,t),x);
 

diff(u(x, t), t) = `+`(`-`(diff(u(x, t), x))) (3)
 

> IC:=u(x,0)=0;
 

u(x, 0) = 0 (4)
 

> N:=200;
 

200 (5)
 

> L:=1;
 

1 (6)
 

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

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

> dydxb:=(u[N+1](t)-u[N](t))/h:
 

> dydx:=1/h*(u[m](t)-u[m-1](t)):
 

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

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

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

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

> eq[0]:=bc1;
 

`+`(u[0](t), `-`(1)) (7)
 

> eq[N+1]:=bc2;
 

diff(u[201](t), t) = `+`(`-`(`/`(`*`(`+`(u[201](t), `-`(u[200](t)))), `*`(h)))) (8)
 

> 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:
 

> u[0](t):=(solve(eq[0],u[0](t)));
 

1 (9)
 

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

> eqs:=[seq(rhs(eq[j]),j=1..N+1)]:
 

> Y:=[seq(u[i](t),i=1..N+1)]:
 

> A:=genmatrix(eqs,Y,'b1'):
 

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

> h:=eval(L/(N+1)):
 

> A:=map(eval,A):
 

> if N > 4 then A:=map(evalf,A);end:
 

> Nrow:=rowdim(A);
 

201 (10)
 

> #exponential(A,t);
 

> #seq([seq(exp(-Nrow*t)*(Nrow*t)^(i-j),j=1..i)],i=1..Nrow);
 

> mat:=Matrix(Nrow,[seq([seq(exp(-Nrow*t)*(Nrow*t)^(i-j)/factorial(i-j),j=1..i)],i=1..Nrow)],shape=triangular[lower]):
 

> Y0:=matrix(N+1,1):for i from 1 to N+1 do Y0[i,1]:=evalf(subs(x=i*h,rhs(IC)));od:evalm(Y0):
 

> Y:=evalm(mat&*(Y0+inverse(A)&*b)-inverse(A)&*b):
 

> #b2:=subs(t=tau,evalm(b)):
 

> #mat2:=subs(t=t-tau,evalm(mat)):
 

> #mat3:=evalm(mat2&*b2):
 

> #mat4:=map(int,mat3,tau=0..t):
 

> #Y:=evalm(mat&*Y0+mat4):
 

> for i from 1 to N+1 do u[i](t):=evalf((Y[i,1]));od:
 

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

> for i from 0 to N+1 do u[i](t):=subs(I=0,u[i](t));od:
 

> setcolors(["Red", "Blue", "LimeGreen", "Goldenrod", "maroon", "DarkTurquoise", "coral", "aquamarine", "magenta", "khaki",  "sienna", "orange", "yellow", "gray"]):
 

> pp:=plot([seq(subs(delta=1,u[i](t)),i=0..N+1,20)],t=0..1,thickness=3);
 

PLOT(CURVES([[0., 1.], [0.217971541666666658e-1, 1.], [0.407627414583333348e-1, 1.], [0.620915279166666667e-1, 1.], [0.835616954166666648e-1, 1.], [.104929818958333324, 1.], [.124740804791666660, 1.],... (11)
 

> pt:=textplot([seq([0.05+i/20*0.09,evalf(subs(delta=1,t=0.05+i/20*0.09,u[i](t))),typeset(u[i]),align={below,right}],i=0..N+1,20)]);
 

PLOT(TEXT([0.5e-1, 1.], _TYPESET(u[0]), ALIGNRIGHT, ALIGNBELOW), TEXT([.1400000000, .9546308230], _TYPESET(u[20]), ALIGNRIGHT, ALIGNBELOW), TEXT([.2300000000, .8389593086], _TYPESET(u[40]), ALIGNRIGHT... (12)
 

> display([pp,pt],title="Figure Exp. 5.34.",axes=boxed,labels=[t,"u"]);
 

Plot_2d
 

> for i from 0 to N+1 do p[i]:=plot(subs(delta=0.1,u[i](t)),t=0..0.4,thickness=3);od:
 

> tf:=1;
 

1 (13)
 

> M:=30;
 

30 (14)
 

> T1:=[seq(tf*i/M,i=0..M)]:
 

> PP:=matrix(N+2,M+1);
 

array( 1 .. 202, 1 .. 31, [ ] ) (15)
 

> 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],subs(delta=1,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,title="Figure Exp. 5.35.",axes=boxed, labels=[x,t,u],orientation=[-45,45]);
 

Plot
 

>