Help with simplifying a 2nd order pde

  • #1
I was given the equation
dp/ds = 4 + 1/e*d/de(e*dp/de)

The derivatives in the equation are partial derivatives

the values of p,s,e are dimensionless numbers.

I am to assume that the solution is separable and then use finite difference method to solve for p, the finite difference method is not a problem. This is where i am having problems. What will the equation be after the assumption is made and the equation is simplified.

I have attempted the question:

I equated the right hand side = 0:

4 + 1/e*d/de(e*dp/de) = 0

and made the partial derivatives total derivatives and then applied the chain rule:

4 + 1/e*dp/de + d/de(dp/de). Is this correct????????
 

Answers and Replies

  • #2
Hootenanny
Staff Emeritus
Science Advisor
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I was given the equation
dp/ds = 4 + 1/e*d/de(e*dp/de)

The derivatives in the equation are partial derivatives

the values of p,s,e are dimensionless numbers.

I am to assume that the solution is separable and then use finite difference method to solve for p, the finite difference method is not a problem. This is where i am having problems. What will the equation be after the assumption is made and the equation is simplified.

I have attempted the question:

I equated the right hand side = 0:

4 + 1/e*d/de(e*dp/de) = 0

and made the partial derivatives total derivatives and then applied the chain rule:

4 + 1/e*dp/de + d/de(dp/de). Is this correct????????
I'm not sure why you have set the right hand side to zero. If you have a separable solution, this will not be the case for non-trivial solutions. The method of separation of variables should be applied as follows.

Let [itex]p(e,s) = q(e)r(s)[/itex]. Then we have

[tex]\frac{\partial}{\partial s}q(e)r(s) = 4 + \frac{1}{e}\frac{\partial}{\partial e}\left(e\frac{\partial}{\partial e} q(e)r(s)\right)\;.[/tex]

Can you take it from here?
 
  • #3
I have attempted it again:

using the equation given i ended up with:

q*dr/ds = 4 + r(1/e*dq/de+d/de(dq/de))

what happens to the 4 then??
 
  • #4
Hootenanny
Staff Emeritus
Science Advisor
Gold Member
9,621
7
I have attempted it again:

using the equation given i ended up with:

q*dr/ds = 4 + r(1/e*dq/de+d/de(dq/de))

what happens to the 4 then??
Let's re-write it so that we can see what's happening more clearly:

[tex]q(e)r^\prime(s) = 4 + r(s)\left[\frac{q^\prime(e)}{e} + q^{\prime\prime}(e)\right][/tex]

This is still a separable equation, the fact that you have an additional constant term doesn't matter. If you're struggling to see it, let [itex]R(s) = 4+r(s)[/itex]. Then [itex]R^\prime(s) = r^\prime(s)[/itex] leading to

[tex]q(e)R^\prime(s) = R(s)\left[\frac{q^\prime(e)}{e} + q^{\prime\prime}(e)\right][/tex]

Can you now continue?
 
  • #5
Yes thank you very much
 

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