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These are replies submitted by KIRAN SAJJAN

@acer all cf columns together and nu columns together sir

[|Gr=0.1, eta= 0.001,0.002..220, cf, nu |,|Gr=0.5, eta= 0.001,0.002..220, cf, nu |,|Gr=1, eta= 0.001,0.002..220, cf, nu |] in this manner.

Fistly for one value of Gr like Gr=0.1 , all corresponding values of cf and nu for eta= (0..20,0.01)

In next columns for the value of Gr=0.5 all corresponding values. In the similar way remaing values.



with(PDEtools); with(plots)


inf := 20


Pr := .71; gamma := .3; Ec := .4



OdeSys := {(diff(Theta(xi, eta), eta, eta))/Pr-(diff(Theta(xi, eta), xi))-gamma*(diff(Theta(xi, eta), eta))+Ec*(diff(u(xi, eta), eta))*(diff(u(xi, eta), eta)) = 0, diff(u(xi, eta), eta, eta)-gamma*(diff(u(xi, eta), eta))-(diff(u(xi, eta), xi))+Gr*Theta(xi, eta) = 0}

Cond := {Theta(0, eta) = 0, Theta(xi, 0) = 1, Theta(xi, inf) = 0, u(0, eta) = 0, u(xi, 0) = 1, u(xi, inf) = 0}


colour := [red, green, blue, black]

GrVals := [.1, .5, 1, 1.5]


for j to numelems(GrVals) do Ans[j] := pdsolve((eval([OdeSys, Cond], Gr = GrVals[j]))[], numeric, time = xi, spacestep = 0.25e-1, timestep = 1) end do


plotA := plots:-display(seq(Ans[k]:-plot(u(xi, eta), xi = 1, eta = 0 .. 12, color = colour[k]), k = 1 .. nops(GrVals)), linestyle = "solid", thickness = 2, labels = ["eta", 'u(xi, eta)'], 'axes' = 'boxed')


plotB := plots:-display(seq(Ans[k]:-plot(Theta(xi, eta), xi = 1, eta = 0 .. 12, color = colour[k]), k = 1 .. nops(GrVals)), linestyle = "solid", thickness = 2, labels = ["eta", 'Theta(xi, eta)'], 'axes' = 'boxed')


plots:-display([plotA], size = [900, 500])

plots:-display([plotB], size = [900, 500])


FU := proc (Y, T, j) if not [Y, T, j]::(list(numeric)) then return ('procname')(args) end if; eval(u(xi, eta), (Ans[j]:-value(xi = T))(Y)) end proc



TTheta := proc (Y, T, j) if not [Y, T, j]::(list(numeric)) then return ('procname')(args) end if; eval(Theta(xi, eta), (Ans[j]:-value(xi = T))(Y)) end proc






interface(rtablesize = 100); interface(displayprecision = 8); Matrix([[Cf, Nux], seq([evalf((D[1](FU))([0.1e-1 .. 7, .1], 1, k)), -evalf((D[1](TTheta))([0.1e-1 .. 7, .1], 1, k))], k = 1 .. nops(GrVals))]); interface(rtablesize = 10); interface(displayprecision = -1)

Matrix(%id = 18446745890973289158)







Dear sir,

 Here i m attaching a similar post  Here i want similar type of answers 

in the given post, the system of Odes, but in my case, i have Pdes. How do I extract similar data for my problem? 


Try to produce one - one values of Kr by using the above answer and combine indivisual plots. I have also did like that only for this type of plots. 

May be for animation plots looping will be work or not i guess.

If it would have worked the experts would have replied for your post.


Thank you so much it is working. 

But for all values of M

how to find numerical values of diff(h(Theta,Theta) and diff(f(Theta,Theta) at Theta= Pi/4


Thank you for your answer

Actually, my problem is three-point boundary

f(-(1/4)*Pi) = 0, g(-(1/4)*Pi) = 0, h(-(1/4)*Pi) = 1, H(-(1/4)*Pi) = 1, f(0) = 1, h((1/4)*Pi) = 1, f((1/4)*Pi) = 0, g((1/4)*Pi) = 0, H((1/4)*Pi) = 1

please look at the pdf  which type of plots i need. new_paper_two_phase.pdf

i want to draw similar plots

starting question i have seperated the boundary i have plotted the graphs and merged both but i m not getting similar graphs there is variations in the answers and plots.

i m giving the reference solved in matlab the boundary conditions are specified in PDF



I have changed the boundary then graphs are comming but in base paper graphs having two phases -1 to 1 range


This is the example i want to solve


How to draw Qz1a[j] versus gamma. At Y=0.5

I can able to draw 

Qz1a[j] versus Y for any value of gamma but reverse i am not able to draw.

Please hlep me to draw with other parameters also i will check once.

Thank you.


cols := [red, blue, black,green]:
 plotAA:= display
  ( [ seq
      ( odeplot
        ( Ans1[k],[gamma,Qz1a[k]],
    'axes'= 'boxed','linestyle' = 'solid',labels=[gamma,'Q'],size=[900,700]
if i kept like this getting error not able to evaluate


Thank you for your responce sir,

I have another  question

for any value of Y i.e, Y=0.5,

How to write plot commond for Qz1a[j] versus gamma( range of gama from 0 to 1) for different values of Q1

And Thetabz1a[j] versus gamma.


substituted all the equations, then also no change in the final expressions. 

If any related questions please tag i will check how to impliment they implimented

I have some more equation in the worksheet how to evaluate Q and Thetab terms.

can we substitute P1=(D1*ga+C1)/(1+ga) and P2= (D1-C1)/(1+ga), Z1= -(N*(C1-D1)^2 )/(1+ga)^2,

because the solution is getting long. to reduce the length of equation by substitution with appendixed term


I have checked with those posts sir.

There is no answers for this type of questions.

Thank you



How u got that plot directly. I m not getting sir

There sre two different  cases in one it is f(eta) and another is f1(eta)

All equation are ode equations after evaluation of f(eta)

Product with x values   for that need the 3d for x=0..1 and eta =0..10

In this it is different  for two different parameters nussueslt number plot i need. In work shhet not showing any error but not getting the plot 

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