## 155 Reputation

11 years, 205 days

## plotting error ...

Maple

Any one help me  to remove the error.

I want to plot the curve for different values of alpha.

here is my codes.

restart:
with(linalg):with(plots):
ge[1]:=diff(u[1](x,t),t)=alpha*diff((u[2](x,t)-1)*diff(u[1](x,t),x),x)+(16*x*t-2*t-16*(u[2](x,t)-1))*(u[1](x,t)-1)+10*x*exp(-4*x):
ge[2]:=diff(u[2](x,t),t)=diff(u[2](x,t),x\$2)+alpha*diff(u[1](x,t),x)+4*(u[1](x,t)-1)+x^2-2*t-10*t*exp(-4*x):
bc1[1]:=u[1](x,t)-1:
bc1[2]:=u[2](x,t)-1:
bc2[1]:=3*u[1](x,t)+diff(u[1](x,t),x)-3:
bc2[2]:=5*diff(u[2](x,t),x)-evalf(exp(4))*(u[1](x,t)-1):
IC[1]:=u[1](x,0)=1:
IC[2]:=u[2](x,0)=1:
NN:=2:
N:=2:
L:=1:
for i to NN do
dydxf[i]:=1/2*(-u[2,i](t)-3*u[0,i](t)+4*u[1,i](t))/h:
dydxb[i]:=1/2*(u[N-1,i](t)+3*u[N+1,i](t)-4*u[N,i](t))/h:
dydx[i]:=1/2/h*(u[m+1,i](t)-u[m-1,i](t));
d2ydx2[i]:=1/h^2*(u[m-1,i](t)-2*u[m,i](t)+u[m+1,i](t)):od:
for i to NN do bc1[i]:=subs(diff(u[1](x,t),x)=dydxf[1],
diff(u[2](x,t),x)=dydxf[2],u[1](x,t)
=u[0,1](t),u[2](x,t)=u[0,2](t),x=0,bc1[i]):od:
for i to NN do bc2[i]:=subs(diff(u[1](x,t),x)=dydxb[1],
diff(u[2](x,t),x)=dydxb[2],u[1](x,t)
=u[N+1,1](t),u[2](x,t)=u[N+1,2](t),x=L,bc2[i]):od:
for i to NN do eq[0,i]:=bc1[i];eq[N+1,i]:=bc2[i]:od:
for i from 1 to N do eq[i,1]:=diff(u[i,1](t),t)= subs(diff(u[1](x,t),x\$2) =
subs(m=i,d2ydx2[1]),
diff(u[2](x,t),x\$2) = subs(m=i,d2ydx2[2]),diff(u[1](x,t),x) =
subs(m=i,dydx[1]),diff(u[2](x,t),x) = subs(m=i,dydx[2]),u[1](x,t)=u[i,1](t),
u[2](x,t)=u[i,2](t),x=i*h,rhs(ge[1])):od:

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

for i to NN do u[0,i](t):=(solve(eq[0,i],u[0,i](t))):od:

for i to NN do u[N+1,i](t):=(solve(eq[N+1,i],u[N+1,i](t))):od:

h:=L/(N+1):

for i from 1 to N do eq[i,1]:=eval(eq[i,1]):od:
for i from 1 to N do eq[i,2]:=eval(eq[i,2]):od:

eqs:=seq(seq((eq[i,j]),i=1..N),j=1..NN):
Y:=seq(seq(u[i,j](t),i=1..N),j=1..NN):

ICs:=seq(u[i,1](0)=rhs(IC[1]),i=1..N),seq(u[i,2](0)=rhs(IC[2]),i=1..N):

sol:=dsolve({eqs,ICs},{Y},type=numeric,stiff=true,maxfun=1000000,abserr=1e-6,relerr=1e-5,output=listprocedure):

Warning, The use of global variables in numerical ODE problems is deprecated, and will be removed in a future release. Use the 'parameters' argument instead (see ?dsolve,numeric,parameters)
for j to NN do for i to N do U[i,j]:=subs(sol,u[i,j](t)):od:od:

for i to NN do U[0,i]:=subs(u[1,1](t)=U[1,1],u[1,2](t)=U[1,2],
u[2,1](t)=U[2,1],u[2,2](t)=U[2,2],u[0,i](t)):od:
for i to NN do U[N+1,i]:=eval(subs(u[N,1](t)=U[N,1],u[N,2](t)=U[N,2],
u[N-1,1](t)=U[N-1,1],u[N-1,2](t)=U[N-1,2],u[N+1,i](t))):od:
tf:=1.:
M:=30:
T1:=[seq(tf*i/M,i=0..M)]:
PP:=matrix(N+2,M+1):
for i from 1 to N+2 do PP[i,1]:=evalf(subs(x=(i-1)*h,rhs(IC[1]))):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-1,1](t))):od:od:

G1:=[seq([ seq([(i-1)*h,T1[j],PP[i,j]], i=1..N+2)], j=1..M+1)]:
t=0.02:
pars:=[0.1,0.5,1,2,5];
clr:=[black,red,green,gold,blue];
for m from 1 to 5 do
G1[m]:=plot([seq(subs(alpha=pars[m],G1[i])],i=0..N+1)],thickness=3,color=clr[j]):od:

display({seq(G1[i],i=1..5)},title="Figure ",axes=boxed,labels=[x,u]);
restart:

## crank nicolson system of difference sche...

Maple

Hai

Help required to plot the graphs for system  difference schemes .

I am attaching the codes and sample graphs and  maple query but still getting  error.

here is the codes

restart;
restart; Digits := 1;
Pr:=0.01:E:=1:a:=0:N:=10:

`&Delta;t`:=0.01:`&Delta;y`:=0.01:

#Discritization scheme

for i from 1 by 1 while i<=N do;
end:
for j from 0 by 1 while j<=N do;
end:

eq1[i, j] := (U[i, j+1]-U[i, j])/`&Delta;t` = (1/2)*Gr*(theta[i, j+1]+theta[i, j])+(1/2)*Gc*(C[i, j+1]+C[i, j])+(U[i-1, j+1]-2.*U[i, j+1]-2.*U[i, j]+U[i+1, j])/(2.*`&Delta;y`)^2-(1/2)*M*(U[i, j+1]+U[i, j]):

eq2[i, j] := (theta[i, j+1]-theta[i, j])/`&Delta;t` = (1/Pr)*(theta[i-1, j+1]-2*theta[i, j+1]+theta[i+1, j+1]+theta[i-1, j]-2*theta[i,j]+theta[i+1,j])/(2.*`&Delta;y`)^2-E*((1/`&Delta;y`)^2*(U[i+1, j]-U[i, j])^2):

eq3[i, j] := (C[i, j+1]-C[i, j])/`&Delta;t` = (1/Sc)*(C[i-1, j+1]-2*C[i, j+1]+C[i+1, j+1]+C[i-1, j]-2*C[i,j]+C[i+1,j])/(2*`&Delta;y`)^2-(K/2)*(C[i, j+1]-C[i, j]):
end do;
Error, reserved word `end` unexpected
end do:
Error, reserved word `end` unexpected

# initial conditions
U[i, 0] := 0:
theta[i, 0]:= 0:
C[i, 0] := 0:

NULL;
U[0,j]:=exp(a*j*`&Delta;t`):
theta[0,j]:=(j*`&Delta;t`):
C[0,j]:=j*`&Delta;t`:

U[N,j]:=0:
theta[N,j]:=0:
C[N,j]:=0:

sys := ([seq])(seq(eq[i, j], j = 0 .. N), i = 1 .. N):
nops(sys);
vars:=indets(sys):
nn := Matrix(N+1, N+1,(i, j)-> U[i-1, j-1]):
##
p:=proc(kk) local U_res,A;
U_res:=solve(eval(sys,k=kk),vars);
A:=eval(nn,U_res);
plots:-matrixplot(A)
end proc;

plots:-(U,[M],Y=0..4);

https://www.mapleprimes.com/questions/225117-How-To-Solve-This-Error-In-Maple

## Galerkin Finite Element Method using map...

Maple

Dear maple user  any one suggest me how to solve  second order coupled differential equation using galerkin finite element method for 8 elements and 10 elements using maple codes

## Different Memory and time for running s...

Maple

whats wrong with the codes while running the codes in maple 13 it will take memory and time as 41.80M, 9.29s while the same code is running in maple 18 it will take 1492.38M , 911.79s

Why the same codes take different time and memory. The codes are here

restart:
Digits:=15:
d1:=0.2:d2:=0.6:L1:=0.2:L2:=0.2:F:=0.3:Br:=0.3:
Gr:=0.2: Nb:=0.1:Nt:=0.3:B:=1:B1:=0.7:m:=1:k:=0.1:
Ro:=1:R1:=1:q:=1:alpha:=Pi/4:
h:=z->piecewise( z<=d1,    1,
z<=d1+L1,   1-(gamma1/(2*Ro))*(1 + cos(2*(Pi/L1)*(z - d1 - L1/2))),
z<=B1-L2/2,  1 ,
z<=B1,  1-(gamma2/(2*Ro))*(1 + cos(2*(Pi/L2)*(z - B1))),
z<=B1+L2/2,  R1-(gamma2/(2*Ro))*(1 + cos(2*(Pi/L2)*(z - B1))),
z<=B,    R1):
A:=(-m^2/4)-(1/4*k):
S1:=(h(z)^2)/4*A-ln(A*h(z)^2+1)*(1+h(z)^2)/4*A:
a2:=Int((1/S1),z=0..1):
b2:=Int((sin(alpha)/F),z=0..1):
c2:=(1/S1)*(-h(z)^6/(6912*A)-h(z)^4/(9216*A)+h(z)^2/(4608*A^3)+ln(1+A*h(z)^2)*(h(z)^6/(576*A)+h(z)^4/(512*A^2)-1/(4608*A^4))):
c3:=Int(c2,z=0..1):
c4:=2*Gr*(Nb-Nt)*c3:
e2:=(1/S1)*(-7*h(z)^4/(256*A)-h(z)^2/(128*A^2)+ln(1+A*h(z)^2)*(3*h(z)^4/(128*A)+h(z)^2/(32*A^2)+1/(128*A^3))):
e3:=Int(e2,z=0..1):
e4:=2*(Nt/Nb)*Br*e3:
l1:=-a2:
l2:=-b2-c4+e4:
Dp:=q*l1+l2:

igRe:=subsindets(Dp,specfunc(anything,Int),
u->Int(Re(op(1,u)),op(2,u),
method=_d01ajc,epsilon=1e-6)):

plot([seq(eval(igRe,gamma2=j),j=[0,0.02,0.06])],gamma1=0.02..0.1,
legend = [gamma2 = 0.0,gamma2 = 0.02,gamma2 = 0.04],
linestyle = [solid,dash,dot],
color = [black,black,black],
labels=[gamma1,'Re(Dp)'],
gridlines=false, axes=boxed);

igIm:=subsindets(Dp,specfunc(anything,Int),
u->Int(Im(op(1,u)),op(2,u),
method=_d01ajc,epsilon=1e-6)):

plot([seq(eval(igIm,gamma2=j),j=[0,0.02,0.06])],gamma1=0.02..0.1,
legend = [gamma2 = 0.0,gamma2 = 0.02,gamma2 = 0.04],
linestyle = [solid,dash,dot],
color = [black,black,black],
labels=[gamma1,'Im(Dp)'],
gridlines=false, axes=boxed);

## Unable to plot the graphs...

Maple 13

Hi, all i am unable to plot the graphs ,can any one help me to overcome the error in plotting the graphs.I am using the maple 13. I am attaching the codes

restart:
with(plots):
with(IntegrationTools):
d1:=0.2:L1:=0.2:L2:=0.2:B1:=0.7:B:=1:beta:=0.01:
d2:=0.6:m:=0.1:k:=0.1:

h:=z->piecewise( z<=d1,    1,
z<=d1+L1,   1-(gamma1/(2))*(1 + cos(2*(Pi/L1)*(z-d1-L1/2))),
z<=B1-L2/2,  1 ,
z<=B1,  1-(gamma2/(2))*(1 + cos(2*(Pi/L2)*(z - B1))),
z<=B1+L2/2,  1-(gamma2/(2))*(1 + cos(2*(Pi/L2)*(z - B1))),
z<=B,    1):

A:=(-m^2/4)-(1/(4*k)):
S1:=(h(z)^2)/(4*A)-ln(A*h(z)^2+1)*(1+h(z)^2)/(4*A):
b1:=evalf((1/S1)):
c1:=evalf(Int(b1,z=0..1)):

plot([seq(eval(c1,gamma2=j),j in[0,0.02,0.06])],gamma1=0.02..0.1,legend = [gamma2 = 0.0, gamma2 = 0.02,gamma2 =0.04],linestyle = [solid,dash,dot],color = [black, black,black],axes=boxed);

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