how to solve this FDM showing some error 
FDM_for_plum_and_sphere_region.mw
please give me solution i waiting for reply

It is not a duplicate question. 

I have given my equation but for the given equation I am not able to solve due to that I have given here if it would have ode. 

I would have implemented the dsolve method to solve but for the pde I can't due to that I have posted the same 

Thank you 

@sand15 

the problem involves two separate regions: Region 1 (the boundary layer around the sphere) and Region 3 (the plume region above the sphere). The governing equations and boundary conditions are different in each region, and they are coupled through matching conditions.

I want to solve for region 1 equations are 14 to 18 and for region 3 equations are 37-41 seperatly, to get table values and plots  for region 1 table 5  values and fig 2 similarly for region 3 table 8  and fig7 need to get.
i have implimented for one one region seperately 

@dharr 
Dear Sir here is the reply for your question

I want to calculate the derivatives in the table  X and Z values,  are at (X = 0.1 and Z = 0). 

Variation of (a) Skin friction ∂W/∂Z​, (b) Heat Transfer ∂θ/∂Z​, and (c) Mass Transfer ∂ϕ/∂Z​ for γ=10.0, Pr=7.0, ε=1.0, Nt=0.4, and Nb=0.2.

plum_model_work

@sand15 
Dear sir,
thank you for your reply 

What I am expecting  it is mentiond in the work sheet.

you have given correct answer, 

i want to check with different parameters also  please specify how to do that in the first worksheet which is given below in that i am getting some wrong plot 
please tell me how to rectify it
pulatile_flow_extended_work1.mw

@sand15 

Please check, sir. I have given published work pdf and worksheet published_flow_work.pdf

peristatic_flow_published_work.mw

I am expecting like fig10 plot. it is converging for the given parameter.

@sand15 

It is general plot sir.

From other paper I given that plot for my work in which range it will converge that range you can consider sir.  According to to parameters effect I will modify it.

It is already published work. In that they have not mentioned x and y range.

@sand15 

it is converging for the below worksheet for the same parameter values pulatile_flow.mw

@sand15 

you are alsolutely correct sir but i am asking that the contour lines plot between the given boundary, similar to the above figure

@Suryakanth 

You are right.

The flow pattern is wrong according to that plot command. 

I think you are expecting this Expected worksheet using this you can implement the same for your work in that only one ode equation.

But in your case you are having 4 different coupled equation. With boundary conditions. 

You are expecting the flow pattern of stream function as u=∂ψ/∂y. I think your are calculating for countour of ∂ψ/∂y from equation 1 as from the worksheet.

I think you need to follow the same method I have mentioned here. 

@Suryakanth 

This command is used for pde only. 

plots[animate](plot, [subs(t=i,rhs(sol)), x = -10 .. 10], i = 0 .. 5,frames=100)

@Suryakanth 

 results1 := []:
results2 := []:
for j from 1 to nops(BrVals) do
    # Step 1: Update ODE systems for this Br
    odesys[1] := eval(OdeSys[1], {R=0.6, Br=BrVals[j]}):
    odesys[2] := eval(OdeSys[2], {R=0.6, Br=BrVals[j]}):
    
    # Step 2: Find A and B using optimization
    ReturnSolutions := 1:
    opt := Optimization:-NLPSolve(TwoDomainsSolver, -1..1, -1..1):
    AB := op(convert(opt[2][1..2], list)):
    
    # Step 3: Solve both regions
    ReturnSolutions := 2:
    S[1], S[2] := TwoDomainsSolver(AB):
    
    # Step 4: Evaluate the four derivatives correctly
    du1_at_m1  := (S[1](-1)):
    du2_at_p1  := (S[2](1)):
    
    
    
    # Step 5: Store results
    results1 := [op(results1), 
                Br=BrVals[j], du1_at_m1
                 ];
results2 := [op(results2), 
                Br=BrVals[j],  du2_at_p1
                 ];
od:

# Step 6: View results

results1;
results2;
from this i am getting table values but there is small clarification i want 

how to extract only derivatives of u[1] ,theta[1]  and u[2] ,theta[2]  instard of getting all values 

in the matrix form i need to extract the data 

@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.