## 155 Reputation

11 years, 205 days

## System of pdes analyically unable to ...

Maple

Dear maple  whats wrong with the code that  maple cannot solve analytically pdes with initial conditions

restart:
sys:={diff(u(x, t), t)=0,diff(v(x, t), t)=0};
IBC:={u(x, 0)=exp((x))/(1+exp((0.5*x)))^2,v(x, 0)=1/(1+exp((0.5*x)))};
pdsolve(sys);
pdsolve(sys,IBC);

## How to generate the graphs for the follo...

Maple

Dear maple user i am facing difficulty to plot the graph   for different values  of parameter M=2,4  and fixing t=j=0 to 2 and   y=i=0 to 4 on x axis and U on y axis. I am unable to plot 2D . I am enclosing the codes and sample graphs.

restart;
# Parameter values:
Pr:=0.71:E:=1:A:=0:Sc:=0.02: K:=1:

a := 0: b := 1: N := 9:
h := (b-a)/(N+1): k := (b-a)/(N+1):

lambda:= 1/h^2:  lambda1:= 1/k^2:
# Initial conditions
for i from 0 to N do
U[i, 0] := h*i+1:
end do:

for i from 0 to N do
T[i, 0] := h*i+1:
end do:
for i from 0 to N do
C[i, 0] := h*i+1:
end do:

# Boundary conditions
for j from 0 to N+1 do
U[0, j] := exp(A*j*lambda);
U[N+1, j] := 0;
T[0, j] := j*lambda1;
T[N+1, j] := 0;
C[0, j] := j*lambda1;
C[N+1, j] := 0
end do:

#Discretization Scheme
for i to N do
for j from 0 to N do
eq1[i, j]:= lambda1*(U[i, j+1]-U[i, j]) = (Gr/2)*(T[i, j+1]+T[i,j])+(Gr/2)*(C[i, j+1]+C[i,j])+(lambda^2/2)*(U[i-1,j+1]-2*U[i,j+1]+U[i+1,j+1]+U[i-1,j]-2*U[i,j]+U[i+1,j])-(M/2)*(U[i,j+1]+U[i,j]) ;
eq2[i, j]:= lambda1*(T[i, j+1]-T[i, j]) = (1/Pr)*(lambda^2/2)*(T[i,j+1]-2*T[i,j+1]+T[i+1,j+1]+T[i-1,j]-2*T[i,j]+T[i+1,j])+(E*lambda^2)*((U[i+1,j]-U[i,j])^2);
eq3[i, j]:= lambda1*(C[i, j+1]-C[i, j]) = (1/Sc)*(lambda^2/2)*(C[i,j+1]-2*C[i,j+1]+C[i+1,j+1]+C[i-1,j]-2*C[i,j]+C[i+1,j])+(K/2)*((C[i,j+1]+C[i,j]))
end do
end do:

#
# Determine the unknowns in the system
#
`union`(  seq(seq( indets( eq1[i,j], name), i=1..N), j=0..N),
seq(seq( indets( eq2[i,j], name), i=1..N), j=0..N),
seq(seq( indets( eq3[i,j], name), i=1..N), j=0..N)
);
#
# And how many unknowns
#
numelems(%);
#
# And the number of equations
#
numelems(eq1)+numelems(eq2)+numelems(eq3);

#
# So if one supplies values for 'Gr' and 'M', and
# (assuming the equations are consistent), one ought
# to be able to get values for C[1..9, 1..10],
# T[1..9,1..10], and U[1..9,1..10]
#
# As an example below, choos Gr=1.0 and M=2, then the
# following obtains a 'solution` afer a minute or so
#
fsolve( eval( [ seq(seq(eq1[i,j], i=1..N),j=0..N),
seq(seq(eq2[i,j], i=1..N),j=0..N),
seq(seq(eq3[i,j], i=1..N),j=0..N)
],
[Gr=1.0, M=2]
)
);

## Error in Analytical solution for differ...

Maple

Hellow maple users, I am getting an error while solving system of differential equations analytically. Please help to recify the error. Thanks in advance. Here is my codes;

restart:
with(DETools):
# S, N  are constant
Eq1:=diff(u(y),y,y)-u(y)=C(y):
Eq2:=diff(T(y),y,y)=u(y)-diff(u(y),y)^2-u(y)^2+S*T(y)+N*T(y):
Eq3:=diff(C(y),y,y)-C(y)=0:
desys:={Eq1,Eq2,Eq3};ics:={u(0)=0,D(u)(0)=h,T(1)=h,D(T)(0)=0,C(1)=h,D(C)(0)=0}:
combine(dsolve(desys union ics,{u(y),T(y),C(y)}));

## 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 18

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);

 1 2 3 4 5 6 7 Last Page 2 of 15
﻿