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. 

@Rouben Rostamian  

Dear sir the parameter values are given in the image above the graph. And the related pdf is given below once check it sir. What are the points you are asking may included in pdf I think. pde_paper_.pdf

Please give the maple code how to implement it .

It is mentioned as homework.

 I am ready to type the code . How to implement for pde I m not getting. Due to that I have posted image file. 

Please update the maple code to solve .

 It will be helpful for me to generate the solution. 

After long back I have posted the question. I have familiar with ode equation. But pde little bit different. 

Please help me to solve these Questions. 

Thank you 

@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.

Download Demo_paper_work.mw

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? 

@SHIVAS 

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.

@mmcdara 

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

@mmcdara 

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 Ode_New_TWO_phase.mw

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

f(±1)=0,g(±1)=0,f(0)=1,h(±1)=1,H(±1)=1

new_paper_two_phase.pdf

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

Ode_New_TWO_phase.mw

@C_R

This is the example i want to solve