tomleslie

Try...

Answered: tomleslie 13428

March 29 2023

0 0

@MANUTTM

op(4, s2)[][];

The function...

Answered: tomleslie 13428

March 24 2023

0 0

'g2' does not contain the name 'q', and so is entirely independent of it.

The expression TC contains the variable names

{Cd, D, O1, O2, P, beta, h1, h2, i, q, r, t1, t2, theta1, theta2}.

However when you differentiate this expression with respect to t2, the result contains only the variables

{Cd, D, O2, beta, h2, i, t2, theta2},

so

{O1, h1, q, t1, theta1}

have been eliminated

fsolve()...

Answered: tomleslie 13428

March 22 2023

1 0

is a numerical solver. Thus every quantity in your equation(s), other than the variable(s) you want fsolve() to compute has to have a numeric value.

I notice that you sayyou are trying to "solve a system of differential equations" for which I would have expected you to be using dsolve(), with, or without the numeric option.

This would be so much easier to diagnose/fix if you uploaded your worksheet using the big green up-arrow in the Mapleprimes toolbar

So why don't you...

Answered: tomleslie 13428

March 20 2023

0 0

just use the built-in help which comes with your Maple release?

As in

Help->What's new

from the menus in the standard GUI

As far as I can tell...

Answered: tomleslie 13428

March 16 2023

0 0

Maple 2017 produces "correct" answer. All subsequent Maple releases produce weird answer.

Using lists rather than expressions sequences appears to make the problem go away in later releases (Maple 2018 - Maple2023)

Can reproduce in Maple2023...

Answered: tomleslie 13428

March 16 2023

0 0

This is a bug, and it is a bad one.

If one uses sequences of names rather than integers, then one can see the difference in the calculations being performed depending on whether the Units() package is loaded or not. The second calculation in the attached makes sense: the first one makes no sense at all.

>

restart;

>	interface(version); with(Units):

Automatically loading the Units[Simple] subpackage

>	A:=a__1, a__2, a__3; B:=b__1, b__2, b__3; B-A

(1)

>

restart;

>	A:=a__1, a__2, a__3; B:=b__1, b__2, b__3; B-A

(2)

>

Download badBug.mw

As I said in my original response...

Answered: tomleslie 13428

March 15 2023

0 0

@delvin

Pretty unlikely that you will be able to solve analytically
Given some additional information, you may be able to solve numerically

HAve you even read...

Answered: tomleslie 13428

March 02 2023

0 0

@MANUTTM

the error message which you are receiving?

Error, (in Optimization:-NLPSolve) could not store 3233180.16916754-HFloat(infinity)*I in a floating-point rtable

I warned you before that generating complex numbers in an optization problem was a very bad idea - so what do think is going to happen when you generate a complex number with an INFINITE imaginary part?

'D'...

Answered: tomleslie 13428

March 02 2023

0 0

is a particularly unfortunate example because, by default, Maple will interpret this as the differential operator, rather than the simple literal 'D'. In consequence it will appear upright rather than in italic, unless you force italic, as in the attached

Download labs2.mw

Again the appearance of the figure on this site is misleading - the way it appears in a worksheet is shown below

As it...

Answered: tomleslie 13428

March 02 2023

0 0

appears in a Maple worksheet

Doesn't make...

Answered: tomleslie 13428

February 28 2023

0 0

@lemelinm

muh of a difference - see the attached

Download simpleInt2.mw

This is not...

Answered: tomleslie 13428

February 23 2023

0 0

@KIRAN SAJJAN

a code writing service. You are expected to do most of the work!

See the attached two files which produce very similar results using Maple's buil-in numeric solvers, and the HPM method

>

a2 := (1-p2)*(1-p1+p1*rs1/rf)+p2*rs2/rf

>

a3 := 1+3*((p1*sigma1+p2*sigma2)/`σf`-p1-p2)/(2+(p1*sigma1+p2*sigma2)/((p1+p2)*`σf`)-((p1*sigma1+p2*sigma2)/`σf`-p1-p2))

>

a4 := (1-p2)*(1-p1+p1*rs1*cps1/(rf*cpf))+p2*rs2*cps2/(rf*cpf)

>

a5 := (ks1+2*kf-2*p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2))/((ks1+2*kf+p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))+2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2)))

>

  OdeSys:= diff(U(Y), Y, Y)/(a1*a2) + Theta(Y) + N*Theta(Y)*Theta(Y) - a3*M*M*U(Y)/a2 - kp*kp*U(Y)/(a1*a2),
           a5*diff(Theta(Y), Y, Y)/a4 + Pr*Ec*(diff(U(Y), Y)^2 + U(Y)^2*kp*kp)/(a1*a2);

  MVals:= [0.5, 1.0, 1.5];
  Cond := U(0) = lambda*D(U)(0), Theta(0) = A + g*D(Theta)(0),
          U(1) = 0, Theta(1) = B;

  for j to numelems(MVals) do
      Ans[j]:= dsolve( eval([OdeSys, Cond], M = MVals[j]),
                       numeric,
                       output = listprocedure
                    );
      Theta_b[j]:= evalf( Int(eval(U(Y), Ans[j])(Y)*eval(Theta(Y), Ans[j])(Y), Y = 0 .. 1)
                          /
                          Int(eval(U(Y), Ans[j])(Y), Y = 0 .. 1)
                        );
  end do;

.7732859212*(diff(diff(U(Y), Y), Y))+Theta(Y)+.5*Theta(Y)^2-.7903642147*M^2*U(Y)-0.6959573287e-1*U(Y), 1.281183414*(diff(diff(Theta(Y), Y), Y))+2.783829316*(diff(U(Y), Y))^2+.2505446384*U(Y)^2

(1)

>

  with(plots):
  cols := [red, blue, black, green, cyan];
  display( [ seq( odeplot( Ans[j],
                           [Y, diff(U(Y), Y)],
                           Y = 0 .. 1,
                           color = cols[j]
                         ),
                  j = 1 .. numelems(MVals)
                )
           ],
           title = typeset(diff(U(Y), Y), "vs ", Y),
           legend = [typeset(M = 0, 5), typeset(M = 1), typeset(M = 1.5)]
         );
  display( [ seq( odeplot( Ans[j],
                           [Y, Theta(Y)],
                           Y = 0 .. 1,
                           color = cols[j]
                         ),
                  j = 1 .. numelems(MVals)
                )
           ],
           title = typeset(Theta(y), "vs ", Y),
           legend = [typeset(M = 0, 5), typeset(M = 1), typeset(M = 1.5)]
         );

>

  interface(rtablesize = 100):
  interface(displayprecision = 5):
  Matrix( [ [0, M = 0.5, M = 0.5, M = 1, M = 1, M = 1.5, M = 1.5],
            [Y, Cf, Nu, Cf, Nu, Cf, Nu],
            seq( [ j,
                   seq( [ eval(diff(U(Y), Y)/a1, Ans[k])(j),
                          eval(-diff(Theta(Y), Y)/(Theta_b[k]*a5), Ans[k])(j)
                        ][],
                        k = 1 .. numelems(MVals)
                      )
                 ],
                 j = 0 .. 1, 0.1
               )
          ]
        );
  interface(rtablesize = 10):
  interface(displayprecision = -1):

(2)

>

Download more3.mw

>

restart:

>

(1)

>

(2)

>

a2 := (1-p2)*(1-p1+p1*rs1/rf)+p2*rs2/rf

>

a3 := 1+3*((p1*sigma1+p2*sigma2)/`σf`-p1-p2)/(2+(p1*sigma1+p2*sigma2)/((p1+p2)*`σf`)-((p1*sigma1+p2*sigma2)/`σf`-p1-p2))

>

a4 := (1-p2)*(1-p1+p1*rs1*cps1/(rf*cpf))+p2*rs2*cps2/(rf*cpf)

>

a5 := (ks1+2*kf-2*p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2))/((ks1+2*kf+p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))+2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2)))

>

Z := 4:

>	U(Y):= add((p^(i))*u[i](Y), i = 0 .. Z):

>	Theta(Y):= add((p^(i))*theta[i](Y), i = 0 .. Z):

>

    odeSys:= { (1-p)*(diff(U(Y),Y,Y))/(a1*a2)
              +
              p*((diff(U(Y),Y,Y))/(a1*a2)+Theta(Y)+N*(Theta(Y)*Theta(Y))-a3*(M*M)*U(Y)/a2-(kp*kp)*U(Y)/(a1*a2)),

              (1-p)*a5*(diff(Theta(Y),Y,Y))/a4
              +
              p*(a5*(diff(Theta(Y),Y,Y))/a4+Pr*Ec*((diff(U(Y),Y))*(diff(U(Y),Y))+(U(Y)*U(Y))*(kp*kp))/(a1*a2))
            }:

   MVals:=[0.5,1.0,1.5]:

   for j from 1 by 1 to numelems(MVals) do

       cond:= u[0](0) = lambda*(D(u[0]))(0), theta[0](0) = A+g*(D(theta[0]))(0),
              u[0](1) = 0, theta[0](1) = B:
       ans[j]:= {}:

       for k from 0 by 1 to Z do
           ans[j]:= evalf( `union`
                           ( ans[j],
                             dsolve
                             ( { eval
                                 ( coeff~(eval(odeSys, M=MVals[j]), p, k),
                                   ans[j]
                                 )[],
                                 cond
                               }
                             )
                           )
                        ):
           cond:= u[k+1](0) =lambda*(D(u[k+1]))(0), theta[k+1](0) = 0, u[k+1](1) = 0, theta[k+1](1) = 0:
       od:

       Theta_b[j]:= int( eval(U(Y), [ans[j][], p=1])*eval(Theta(Y), [ans[j][], p=1]), Y=0..1)
                    /
                    int( eval(U(Y), [ans[j][], p=1]), Y=0..1):
   od:

>

  with(plots):
  cols := [red, blue, black, green, cyan]:

  display
  ( [ seq
      ( plot
        ( diff(eval( U(Y), [ans[j][], p=1]),Y),
          Y=0..1,
          color=cols[j]
        ),
        j=1..numelems(MVals)
      )
    ],
    title = "diff(U(Y), Y) vs Y)",
    legend = [typeset(M = 0.5), typeset(M = 1), typeset(M = 1.5)]
  );
  display
  ( [ seq
      ( plot
        ( eval( Theta(Y), [ans[j][], p=1]),
          Y=0..1,
          color=cols[j]
        ),
        j=1..numelems(MVals)
      )
    ],
    title = "Theta(Y) vs Y)",
    legend = [typeset(M = 0.5), typeset(M = 1), typeset(M = 1.5)]
  );

>

  interface(rtablesize=100):
  interface(displayprecision=5):
  Matrix( [ [ Y, Cf, Nu, Cf, Nu, Cf, Nu],
              seq
              ( [ j,
                  seq( [eval(diff(U(Y),Y)/a1,[ans[k][], Y=j,p=1]),
                        eval(-1/(Theta_b[k])*diff(Theta(Y),Y)/a5,[ans[k][],Y=j,p=1])
                       ][],
                       k=1..numelems(MVals)
                     )

                ],
                j=0..1, 0.1
              )
          ]
      );
  interface(rtablesize=100):
  interface(displayprecision=-1)

(3)

>

Download bhpm3.mw

Ids there no end...

Answered: tomleslie 13428

February 23 2023

0 0

@KIRAN SAJJAN

to what yoiu want?

There are two attached files: one solve the problem using Maple's built-in hunerical solvers, and one using the "HPM method". They produce the following two tables (where I have restricted the display precision (but not the precision used in actual calculation), just to make comparisons easier.

Maple's numeric solvers

HPM method

OK, these are different, but generally they are within about 3%. Since I have absolutely no idea how accurate I would expect a 4-term HPM method to be, such a discrepancy doesn't bother me much

more2.mw

bhpm2.mw

Worksheet...

Answered: tomleslie 13428

February 23 2023

0 0

@KIRAN SAJJAN

with requested graphs and table is attached.

At no time did I enter as input either U(Y)(Y) or Theta(Y)(Y).

In Maple output, this is an artefact of using the option output=listprocedure in the dsolve() command. Maple will give the relevant procedures the names U(Y) and Theta(Y), but these are just literal procedureNames. Applying these literal procedure names to a variable 'Y' is what results in U(Y)(Y) and Theta(Y)(Y)

>

a2 := (1-p2)*(1-p1+p1*rs1/rf)+p2*rs2/rf

>

a3 := 1+3*((p1*sigma1+p2*sigma2)/`σf`-p1-p2)/(2+(p1*sigma1+p2*sigma2)/((p1+p2)*`σf`)-((p1*sigma1+p2*sigma2)/`σf`-p1-p2))

>

a4 := (1-p2)*(1-p1+p1*rs1*cps1/(rf*cpf))+p2*rs2*cps2/(rf*cpf)

>

a5 := (ks1+2*kf-2*p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2))/((ks1+2*kf+p1*(kf-ks1))*(ks2+2*kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))+2*p2*(kf*(ks1+2*kf-2*p1*(kf-ks1))/(ks1+2*kf+p1*(kf-ks1))-ks2)))

>

interface(rtablesize = 100); Matrix([[0, M = .5, M = .5, M = 1, M = 1, M = 1.5, M = 1.5], [Y, Cf, Nu, Cf, Nu, Cf, Nu], seq([j, seq([(eval((diff(U(Y), Y))/a1, Ans[k]))(j), (eval(-(diff(Theta(Y), Y))/(Theta_b[k]*a5), Ans[k]))(j)][], k = 1 .. numelems(MVals))], j = 0 .. 1, .1)]); interface(rtablesize = 100)

Matrix(%id = 36893488148298344076)

(1)

>

Download more.mw

That depends a lot...

Answered: tomleslie 13428

February 22 2023

0 0

@KIRAN SAJJAN

on whether you can define "stream line", "Nussel number" and "skin friction" in terms of the quantities produced in the worksheet.

I can't.

If you can't - then you are stuffed!

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Try...

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fsolve()...

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As far as I can tell...

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As I said in my original response...

HAve you even read...

'D'...

As it...

Doesn't make...

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