@Suryakanth 

Thank you. 

I am getting plots similarly i need table values command how to get the derivative of u1, u2 and theta1 and theta2 

how to get the table values for Differnt Br values in matrix form

I want only table values how to extract it 

it is little bit different one. no one having idea to solve this i have discussed with our friends also.

@sand15 
I have followed your previous suggestions and made the required changes, but unfortunately it is still not working for me. I have been trying for the past 12 hours and also took help from some friends who use Maple, but they also found it difficult to make it run.

When I change the value of j, only one line is plotted; I am not getting all curves together in a single plot. In my earlier worksheet, the conditions I had were different, and with those conditions it is not working now. Even after changing all conditions as suggested, the looping is not working in this worksheet.

Also, the table values are not appearing with the old commands. I sincerely appreciate the time you have taken to reply despite your busy schedule, and I thank you for your support. If I had been able to obtain the plots and table values as expected, I would not have needed to post my question again.

please chek this whorksheet sirmodified_symmetry_work.mw

with(plots):
  cols := [red, blue, black,green]:
plotD:= display
  ( [ seq
      ( odeplot
        ( S[1][k],[y,Theta[2](y)],
          y=0..1,
          color=cols[k]
        ),
        k=1..numelems(BrVals)
      )
    ],
    'axes'= 'boxed',labels=[y,'Theta(y)'],font = [TIMES,BOLD,14], size=[700,700],legend = [typeset(Br = 0.1), typeset(Br = 0.3), typeset(Br = 0.5), typeset(Br = 0.8)], legendstyle = [font = [times, bold, 14]]
  );  

@sand15  

Thank you for your help. 

But i have a dought in the worksheet

You kept Br values as BrVals := [.1, .3, .5, .8]:

but in the plot and table values, i got only one value for Br=0.1

How to get all plots at a time modified_symmetry_work.mw 

in your worksheet you taken all four values but ploted only for one value.

@sand15 
Thank you for your explanation. I understand the approach of introducing parameters at the boundaries and using optimization to satisfy the continuity conditions.

In my case, I would like to solve the two regions separately:

• Region 1: from y=−1y=-1 to y=0y=0

• Region 2: from y=0y=0 to y=1y=1

Then, I want to plot the solutions for each region individually rather than combining them into a single plot. Could you please guide me on how to modify the Maple code so that I get two separate plots—one for Region 1 and one for Region 2—after solving?

similar published work given here similar boundary and two regions 

Entropy generation for mixed convection flow in vertical annulus with two regions hydromagnetic viscous and Cu-Ag water hybrid nanofluid through porous zone: a comparative numerical study - ScienceDirect

@sand15 

There are two different regions are there sir first two equations are region one from -1 to 0 and other is 0 to 1 

I want two seperate plots 

In the paper they given combinly in one plot but i need seperately. 

From these conditions i have solved malualy i have not got the solution. Due to that i have checked here. 

Long back also i have posted similar question regarding two region but they have ploted those graphs in matlab and that equations also pde. 

But in this case it is different. 

I need two different cases 

One is from -1 to 0 and 

Another from 0 to 1 

Seperate plots for solving the ode.

But i am not getting how to take boundaries separately 

For both cases. 

@dharr 

Dear sir, 

According to the suggestions I have checked the values of parameters given in the base pdf. Nusselt number value and shearhood number and not exactly getting same when we exicute the worksheet. 

What is the error in the calculation,?

Why Different reading is getting for the given parameter values. There is lage difference in the values. 

Thank you.

@dharr 

Thank you for your suggestions. Now the worksheet is executing, but for the given parameters, the graphs are not the same as given in the base paper.sachin_base_paper.mw

refence_of_sachin_base_paper.pdf why this error is getting for the given parameter values in plots as well as table values?  

@dharr 

Dear sir 

I have attached the pdf also according to the suggestions I have checked then also am getting the error. Sir

In the worksheet the nusselt number and shearhood number is not matching with base paper given in the pdf 

I have checked the similar papers in that also similar solution implemented as given in the pdf.

@dharr 

is this  correct ? sachin_base_paper.mw  but here I'm getting wrong values for the table which is given in base paper Table_values_of_sachin_base.mw  refence_of_sachin_base_paper.pdf

What is the mistake in the code ?

Why the graph is not converging? For the given value of parameters.

Please Help me to solve the problem.

@Rouben Rostamian  please check this worksheet sir getting some error in boundary  pde_error_two_boundary.mw

@Rouben Rostamian  

restart:

with(PDEtools):

declare(u(r,z), v(r,z), w(r,z), Theta(r,z), p(r,z)):

PDE1 := diff(r*u(r,z), r)/r + diff(w(r,z), z) = 0:

PDE2 := u(r,z)*diff(u(r,z),r) + w(r,z)*diff(u(r,z),z) - v(r,z)^2/r - 2*v(r,z)

         = -diff(p(r,z),r) + (1/lambda^2)*(epsilon^2*(diff(u(r,z),r$2) + diff(u(r,z),r)/r - u(r,z)/r^2) + diff(u(r,z),z$2)):

PDE3 := u(r,z)*diff(v(r,z),r) + w(r,z)*diff(v(r,z),z) + (u(r,z)*v(r,z))/r + 2*u(r,z)

         = (1/lambda^2)*(epsilon^2*(diff(v(r,z),r$2) + diff(v(r,z),r)/r - v(r,z)/r^2) + diff(v(r,z),z$2)):

PDE4 := u(r,z)*diff(w(r,z),r) + w(r,z)*diff(w(r,z),z)

         = -1/(epsilon^2)*diff(p(r,z),z) + (1/lambda^2)*(epsilon^2*(diff(w(r,z),r$2) + diff(w(r,z),r)/r) + diff(w(r,z),z$2)):

PDE5 := u(r,z)*(-Theta(r,z)/r + diff(Theta(r,z),r)) + w(r,z)*diff(Theta(r,z),z)

         = 1/Pr/lambda^2*(epsilon^2*(diff(Theta(r,z),r$2) + diff(Theta(r,z),r)/r - Theta(r,z)/r^2) + diff(Theta(r,z),z$2)):

BCs := {u(r,1) = L*diff(u(r,1),z), u(r,1) = -L*diff(u(r,1),z), 

        v(r,1) = 0, w(r,1) = 0, Theta(r,1) = 1}:

sol := pdsolve({PDE1, PDE2, PDE3, PDE4, PDE5}, {u(r,z), v(r,z), w(r,z), Theta(r,z), p(r,z)},  BCs):

@Rouben Rostamian  

In maple only give me the suggestions how to implement the conditions for pde equation. I don't want to convert pde into ode. I need the solution of pde equation only .

I seen some post of pde solution also those are also nonlinear equation. 

How to get this pde solution without transferring into ode 

@Rouben Rostamian  

They used siries solution in terms of  v0 and v1  but it difficult to use series solution for the complex terms because in my extension I have used multiple parameters .

 I need the solution by pde only because similar articles not used series solution they given directly pde equation. 

Is it chance to simply by pde only ? 

Suggest me any post related to pde equation with two different boundary .

In that they have given -1 to 0 and again 0 to 1 given. 

In matlab also I have tied but for only 0to 1 I'm getting or -1 to 0 only getting. Combining both boundary in single plot not getting.