## 90 Reputation

9 years, 197 days

## How to rectify the error while using H...

Maple

Dear maple user how to rectify the error  in solving the coupled differential equation using homotropy perturbation method and direct differentiations and compare the two result by plotting the graphs :

```restart:
with(PDEtools):
L:=4:Nb:=1:Nt:=1:#k is some constant
f(x):=sum((p^i)*f[i](x),i=0..L):
g(x):=sum((p^i)*g[i](x),i=0..L):
HO1:=(1-p)*(diff(f(x),x,x))+p*(((1/x)*(diff(f(x),x))+Nb*((diff(f(x),x))*diff(g(x),x))+Nt(diff(f(x),x)^2))):
expand(%):
collect(%,p):
HO2:=(1-p)*(diff(g(x),x,x))+p*(((1/x)*(diff(g(x),x))+(Nb/Nt)*((diff(f(x),x,x))+(1/x)*diff(f(x),x)))):
expand(%):
collect(%,p):
HO2:=%:
declare(f(x),g(x),prime=x):
for i from 0 to L+1 do equa[1][i]:=coeff(HO1,p,i)=0 end  do:
for i from 0 to L+1 do equa[2][i]:=coeff(HO2,p,i)=0 end  do:
con[1][0]:=f[0](0)=(h(x)/64),(D(f[0]))(0)=0:
con[2][0]:=g[0](0)=-(k-h(x)^2/4),(D(g[0]))(0)=0:
for j from 1 to L do:
con[1][j]:=f[j](0)=h(x),(D(f[j]))(0)=0:
con[2][j]:=g[j](0)=h(x),(D(g[j]))(0)=0:
end do;
for i from 0 to L do;
dsolve({equa[1][i],con[1][i]},f[i](x));
f[i](x):=rhs(%);
f[i](x):=evalf(%);
dsolve({equa[2][i],con[2][i]},g[i](x));
g[i](x):=rhs(%);
g[i](x):=evalf(%);
end do;
for u from 0 to L-1 do:
f[u](_z1):subs(x=_z1,f[u](x));
g[u](_z1):subs(x=_z1,g[u](x));
f[u+1](x):=value(simplify(f[u+1](x)));
f[u+1](x):=simplify(%);
g[u+1](x):=value(simplify(g[u+1](x)));
g[u+1](x):=simplify(%);
end do:
f(x):=evalf(simplify(sum(f[n](x),n=0..L)));
#### direct calculations
#direct solve the two equations in terms of f(x) and g(x) for h(x)=e^x where Nt and Nb #are parameters and its takes some values example 1,1
restart:
with(DETools):
with(plots):
with(IntegrationTools):
Nb:=1:Nt:=1:h(x):=e^x:
Eq1 := (diff(f(x),x,x))+(((1/x)*(diff(f(x),x))+Nb*((diff(f(x),x))*diff(g(x),x))+Nt(diff(f(x),x)^2))):
Eq2 := (diff(g(x),x,x))+(((1/x)*(diff(g(x),x))+(Nb/Nt)*((diff(f(x),x,x))+(1/x)*diff(f(x),x)))):

Cd1 := f(0) = h(x), (D(f))(0) = 0:
dsys := {Cd1, Eq1}:
dsol := dsolve(dsys, numeric, output = operator):
#dsol(.1):
plots[odeplot](dsol, [x, diff(f(x), x\$1)], 0 .. 5, color = green):
Cd2 := g(0) = h(x), (D(g))(0) = 0:
dsys := {Cd1, Cd2, Eq1, Eq2}:
dsol := dsolve(dsys, numeric, output = operator):
plots[odeplot](dsol, [x, f(x)], 0 .. 5, color = red);
plots[odeplot](dsol, [x,g(x)], 0 .. 5, color = black);

```

## HPM error with different do and end loo...

Maple

Dear maple user,

I have codes for Differential equations while applying one do and end loop i am able to plot the graph of G(x) while same problem with other way of applying do and end loop i am unable to plot. whats wrong with do and end loop. These are codes available in maple primes . while combining i am unable to plot .

any one resolve it.

restart:
with(DETools):
with(plots):
with(IntegrationTools):
de0 := {
(1-p)*(diff(f(x),x,x,x))+p*(diff(f(x),x,x,x)+(1/2)*f(x)*(diff(f(x),x,x))),
(1-p)*(diff(g(x),x\$2))/Pr+p*((diff(g(x),x\$2))/Pr+(1/2)*f(x)*(diff(g(x),x)))}:

ibvc0 := {f(0),(D(f))(0),(D(f))(5)-1,g(0)-1,g(5)}:
n:=2:

F := unapply( add(b[k](x)*p^k,k=0..n), x ):
G := unapply( add(c[k](x)*p^k,k=0..n), x ):

de := map( series, eval( de0, {f=F,g=G} ), p=0, n+1 ):

for k from 0 to n do

if k = 0 then
ibvc := expand( eval[recurse]( ibvc0, {f=F,g=G,p=0} ) ):
else
ibvc := { b[k](0), D(b[k])(0), (D@@2)(b[k])(0), c[k](0), D(c[k])(0) }:
end if:

sys := simplify( map( coeff, de, p, k ) ) union ibvc:
soln := dsolve( sys ):

b[k] := unapply( eval( b[k](x), soln ), x ):
c[k] := unapply( eval( c[k](x), soln ), x ):

end do:

'F(x)' = F(x)+O(p^(n+1)):
'G(x)' = G(x)+O(p^(n+1)):

Pr:=1:
plot(eval(G(x), p = 1), x = 0 .. 5, color = blue):
###### Same problem with other  way of do and and end loop unable to plot with G(x)
restart:
with(DETools):
with(plots):
with(IntegrationTools):
Pr:=1:
de1 := (1-p)*(diff(f(x), `\$`(x, 3)))+p*(diff(f(x), `\$`(x, 3))+(1/2)*f(x)*(diff(f(x), `\$`(x, 2))));
de2 := (1-p)*(diff(g(x), `\$`(x, 2)))/Pr+p*((diff(g(x), `\$`(x, 2)))/Pr+(1/2)*f(x)*(diff(g(x), x)));
ibvc := f(0), (D(f))(0), (D(f))(5)-1, g(0)-1, g(5); n := 2; F := unapply(add(b[k](x)*p^k, k = 0 .. n), x); G := unapply(add(c[k](x)*p^k, k = 0 .. n), x);
DE1 := series(eval(de1, f = F), p = 0, n+1);
DE2 := series(eval(de2, g = G), p = 0, n+1);
CO := map(coeffs, eval([ibvc], f = F), p); CT := map(coeffs, eval([ibvc], g = G), p);

for k from 0 to n do IBVC1 := select(has, C*T, c[k]); slv := dsolve({coeff(DE2, p, k), op(IBVC1)}); c[k] := unapply(rhs(slv), x) end do;
G(x) = G(x)+O(p^(n+1));
plot(eval(G(x), p = 1), x = 0 .. 5);

## Unable to display plots...

Maple

Hellow,

I am unable to combine the graphs. I have a plot structure as

P1:=plots[odeplot](dsol, [x, F(x)], 0 .. 5, color = red);

P2:=plot(eval(F(x), p = 1), x = 0 .. 5, color = blue);

I want to combine two structure and display in same plot

display (P1,P2);

## How to plot real part only in multiple p...

Maple

Help required to plot the real part only in  multiple plots. i have written some codes  please help me to rectify the errors.Thanks in advance

restart:
with(plots):
n:=0.75:Eh:=100:mn:=1:t0:=0.2:
a1:=(mn/t0)^((n-1)/(n))*(tb)^(1/n):a3:=Eh/tb:a4:=(Eh)^2:
U1:=(a1/Eh)*(-1+(a3*r/Eh))^(1/n)*(n*a4/(1+n))*(1/a3-(r/Eh))-a1*(-1+(1/tb))^(1/n)*(n/(n+1)*(tb-1)):
plot([Re(seq(eval(U1,tb=j),j in[0.8,0.9,1.0]))],r=0..1,legend = [tb =0.8, tb=0.9,tb =1.0],  labels = ["z ", "U"], labeldirections = ["horizontal", "vertical"],  linestyle = [solid,dash,dot],color = [black, red,green]);

## Area outside cardiode and inside circ...

Maple

Dear maple user ,

I want to draw the area outside r=2+2*cos(theta) and inside r=3 ,where theta=0..2*pi.

I am attacing the sample figure.