Example 5.9
| > | restart; |
| > | with(linalg):with(plots): |
| > | UseHardwareFloats := true; |
| (1) |
| > | ge:=diff(u(x,t),t)=diff(u(x,t),x$2); |
| (2) |
| > | bc1:=u(x,t); |
| (3) |
| > | bc2:=u(x,t); |
| (4) |
| > | IC:=u(x,0)=1; |
| (5) |
| > | N:=40; |
| (6) |
| > | L:=1; |
| (7) |
| > | 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: |
| > | 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: |
| > | eq[N+1]:=bc2: |
| > | 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))); |
| (8) |
| > | u[N+1](t):=solve(eq[N+1],u[N+1](t)); |
| (9) |
| > | for i from 1 to N do eq[i]:=eval(eq[i]);od: |
| > | eqs:=[seq(rhs(eq[j]),j=1..N)]: |
| > | Y:=[seq(u[i](t),i=1..N)]: |
| > | A:=genmatrix(eqs,Y,'b1'): |
| > | b:=matrix(N,1):for i to N 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: |
| > | #evalm(A); |
| > | Nrow:=rowdim(A): |
| > | l:=evalf(eigenvalues(A)): |
| > | for i to Nrow do lambda[i]:=l[i];od: |
| > | 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]:od: |
| > | for i to Nrow do beta[i]:=fsolve(eqx[i],beta[i]);od: |
| > | XX[k]:=map(eval,evalm(X)): |
| > | for i to Nrow do unassign('beta[i]'):od:od: |
| > | P:=Matrix([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]*t):od: |
| > | mat:=evalm(P&*expD1&*inverse(P)): |
| > | Y0:=matrix(N,1):for i from 1 to N do Y0[i,1]:=evalf(subs(x=i*h,rhs(IC)));od:evalm(Y0): |
| > | 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 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: |
| > | 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+1,5)],t=0..0.4,thickness=3); |
| (10) |
| > | pt:=textplot([seq([0.05+i*0.002,evalf(subs(delta=1,t=0.05+i*0.002,u[i](t))),typeset(u[i]),align=right],i=0..N+1,5)]); |
| (11) |
| > | display([pp,pt],title="Figure Exp.5.30.",axes=boxed,labels=[t,"u"]); |
![]() |
| > | 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:=0.1; |
| (12) |
| > | M:=30; |
| (13) |
| > | T1:=[seq(tf*i/M,i=0..M)]; |
| (14) |
| > | PP:=matrix(N+2,M+1); |
| (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,axes=boxed,title="Figure Exp. 5.31.",labels=[x,t,u],orientation=[45,45]); |
![]() |
| > |