# From PDE to ODE ? + research

From PDE to ODE ?!! + research

## Homework Statement

In the attached research, What are the steps that we work to transform the equation (1) to (8)

(1) and (8)

## The Attempt at a Solution

I know that they used similarity transformations but I do not know how to do it by myself (how to substitute correctly) ...

can you help me to I understand the steps ... then I will do the others In the same way.

Thanks

#### Attachments

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phyzguy

Do you know how to apply the chain rule for partial differentiation? Try going to Wikipedia and searching on "chain rule", and reading the section on several variables. Then I think if you apply this to the research paper you attached you will see how they arrived at the equations. Basically, they chose a new set of variables so that after applying the chain rule and collecting terms, many terms cancelled and the original set of PDEs was converted into a set of ODEs.

Do you know how to apply the chain rule for partial differentiation? Try going to Wikipedia and searching on "chain rule", and reading the section on several variables. Then I think if you apply this to the research paper you attached you will see how they arrived at the equations. Basically, they chose a new set of variables so that after applying the chain rule and collecting terms, many terms cancelled and the original set of PDEs was converted into a set of ODEs.
Thank you for this hint...I will try it :) and read:

http://en.wikipedia.org/wiki/Chain_rule

https://www.physicsforums.com/library.php?do=view_item&itemid=353

Q: How do I choose the appropriate similarity transformations to a particular system?!​

please ... help me ... just the first term in eq(1)

I try but failed :(

phyzguy

Show us your calculations on the first term in Equation 1. Learn how to use TeX so you can type it in.

I will try :(

is it right that u is in 3 variables : x and Fi Dash and M ?

please ... Is this true?!! I took the second term in eq(1)

$$\partial w$$ / $$\partial z$$

= ( $$\partial w$$ / $$\partial \varphi$$ ) . ( $$\partial \varphi$$ / $$\partial \eta$$ ) . ( $$\partial \eta$$ / $$\partial z$$ )

= (- $$\sqrt{b\nu}$$ ) . ( $$\grave{\varphi}$$ ) .( $$\sqrt{b/\nu}$$ )

= -b $$\grave{\varphi}$$​

phyzguy

I think this is correct. Now if you look at the first term in equation 1, you see:

$$\frac{\partial{u}}{\partial{x}} = b \phi'(\eta)$$

so the two terms cancel and equation 1 is automatically satisfied. Keep going!

I think this is correct. Now if you look at the first term in equation 1, you see:

$$\frac{\partial{u}}{\partial{x}} = b \phi'(\eta)$$

so the two terms cancel and equation 1 is automatically satisfied. Keep going!
Thanks ...I appreciate it

but how do I get the equation (8) ?!!

and

is it right that u is in 3 variables : x and Fi Dash and M ?

Last edited:
phyzguy

Yes, you need to consider u to be a function of x, $$\phi'$$, and M. You need to evaluate all of the partial derviatives, like you did for

$$\frac{\partial{w}}{\partial{z}}$$

and plug these into equations 2-5, and you should come out with equations 8-12.

Yes, you need to consider u to be a function of x, $$\phi'$$, and M. You need to evaluate all of the partial derviatives, like you did for

$$\frac{\partial{w}}{\partial{z}}$$

and plug these into equations 2-5, and you should come out with equations 8-12.
ok, but eq (1) say:

$$\frac{\partial{u}}{\partial{x}}$$ + ( $$\partial w$$ / $$\partial z$$) =0

so, I have evaluate all of the partial derviatives but How can I have eq (8) :( I do not understand !!!!!!!

phyzguy

Equation 8 comes from equation 2. Since u has a term proportional to $$\phi'$$, when you evaluate
$$\frac{\partial^2{u}}{\partial{z^2}}$$ , you will get a term in $$\phi'''$$ . Do you see?

Equation 8 comes from equation 2. Since u has a term proportional to $$\phi'$$, when you evaluate
$$\frac{\partial^2{u}}{\partial{z^2}}$$ , you will get a term in $$\phi'''$$ . Do you see?
ok ... I will try :)

Thank you