You are givien the "similarity" variable. It's What you need to do now is substitute the form of your solution into the PDE.
If
then
Your PDE becomes (after dividing by )
.
You must now create IC's for this ODE from your given BCs/ICs.
The original question is:
Try and apply the Similarity solution method to the following boundary value problems for .
for all with boundary conditions
as
for .
I know from my tutorial that I should first find where is an unknown function of similarity variable . What I don't know is how to find the similarity variable and the formula .
Please tell me the procedure for finding similarity solution. Thank you in advance.
You are givien the "similarity" variable. It's What you need to do now is substitute the form of your solution into the PDE.
If
then
Your PDE becomes (after dividing by )
.
You must now create IC's for this ODE from your given BCs/ICs.
Thank you for your help. The question does require finding the ODE of but also requires finding the similarity variable to start with. After getting the solution from my lecturer I typed up the solution below with slight variations according to my understanding.
1. Let , and where are arbitrary and are to be determined.
2. Then ,
and
3. Substituting above partial derivatives into original PDE results in
Because are arbitrary and undertermined, choose .
4. The boundary conditions for are written into boundary condition for as
and
As are arbitrary and undetermined, choose .
5. Solving and results in
6. Substitute above values of and into to get
where is still arbitrary. Let , then
Therefore define similarity variable and function so that
I am still having trouble with another similarity question. I will post it later in this thread.