Transforming PDE to ODE: How Can We Subsitute Correctly?

In summary, the conversation discusses the steps and transformations involved in converting an equation from a partial differential equation (PDE) to an ordinary differential equation (ODE). The use of similarity transformations and the chain rule for partial differentiation are suggested as methods to solve the problem. The conversation also touches on the importance of using TeX in calculations and understanding the variables involved in the equations. Ultimately, the goal is to arrive at Equation 8, which comes from Equation 2 through the evaluation of partial derivatives.
  • #1
thepioneerm
33
0
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)

Homework Equations



(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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  • #2


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 canceled and the original set of PDEs was converted into a set of ODEs.
 
  • #3


phyzguy said:
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 canceled 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
 
  • #4


Q: How do I choose the appropriate similarity transformations to a particular system?!​
 
  • #5


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

I try but failed :(
 
  • #6


Show us your calculations on the first term in Equation 1. Learn how to use TeX so you can type it in.
 
  • #7


I will try :(
 
  • #8


is it right that u is in 3 variables : x and Fi Dash and M ?
 
  • #9


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



[tex]\partial w[/tex] / [tex]\partial z[/tex]

= ( [tex]\partial w[/tex] / [tex]\partial \varphi[/tex] ) . ( [tex]\partial \varphi[/tex] / [tex]\partial \eta[/tex] ) . ( [tex]\partial \eta[/tex] / [tex]\partial z[/tex] )

= (- [tex]\sqrt{b\nu}[/tex] ) . ( [tex]\grave{\varphi}[/tex] ) .( [tex]\sqrt{b/\nu}[/tex] )

= -b [tex]\grave{\varphi}[/tex]​
 
  • #10


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

[tex]\frac{\partial{u}}{\partial{x}} = b \phi'(\eta)[/tex]

so the two terms cancel and equation 1 is automatically satisfied. Keep going!
 
  • #11


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

[tex]\frac{\partial{u}}{\partial{x}} = b \phi'(\eta)[/tex]

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

thepioneerm said:
is it right that u is in 3 variables : x and Fi Dash and M ?
 
Last edited:
  • #12


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

[tex]\frac{\partial{w}}{\partial{z}}[/tex]

and plug these into equations 2-5, and you should come out with equations 8-12.
 
  • #13


phyzguy said:
Yes, you need to consider u to be a function of x, [tex]\phi'[/tex], and M. You need to evaluate all of the partial derviatives, like you did for

[tex]\frac{\partial{w}}{\partial{z}}[/tex]

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

ok, but eq (1) say:

[tex]\frac{\partial{u}}{\partial{x}}[/tex] + ( [tex]\partial w[/tex] / [tex]\partial z[/tex]) =0

so, I have evaluate all of the partial derviatives but How can I have eq (8) :( I do not understand !
 
  • #14


Equation 8 comes from equation 2. Since u has a term proportional to [tex]\phi'[/tex], when you evaluate
[tex]\frac{\partial^2{u}}{\partial{z^2}}[/tex] , you will get a term in [tex]\phi'''[/tex] . Do you see?
 
  • #15


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

ok ... I will try :)

Thank you
 

1. What is the difference between a PDE and an ODE?

A PDE (partial differential equation) involves functions with more than one independent variable, while an ODE (ordinary differential equation) involves functions with only one independent variable. In other words, a PDE deals with functions that have multiple variables, while an ODE deals with functions that have a single variable.

2. What are some real-world applications of PDEs and ODEs?

PDEs are used to model physical phenomena such as heat transfer, fluid dynamics, and quantum mechanics. ODEs are used to model motion, growth, and decay in various systems, such as population dynamics, chemical reactions, and electrical circuits.

3. What are some methods for solving PDEs and ODEs?

There are numerous methods for solving PDEs and ODEs, including analytical methods (such as separation of variables and Laplace transforms), numerical methods (such as finite difference and finite element methods), and computer simulations.

4. How does research in PDEs and ODEs contribute to other fields of science?

Research in PDEs and ODEs has wide-ranging applications in fields such as physics, engineering, biology, economics, and finance. By providing models and solutions for complex systems, this research helps us understand and predict real-world phenomena and make informed decisions.

5. What are some challenges in solving PDEs and ODEs?

One of the main challenges in solving PDEs and ODEs is finding analytical solutions, as many equations are too complex to be solved by hand. Another challenge is the accuracy of numerical methods, as small errors can accumulate and lead to significant discrepancies in the final solution. Additionally, understanding the physical or mathematical meaning of the solutions can also be a challenge.

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