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Simple Substituting and Rearranging

Caccioppoli

New member
Apr 19, 2013
5




Hello,

may someone be so kind to explain how to arrive, step by step, from equation 23 to 28?

Most of all I would like to understand the approximation with delta: if I substitute eq26 in 25 I get a different result (e.g. delta^3 terms).

See the attached image.

Thank you very much.

PS
eq24 may be taken as it is, I mean, phi is simply "(A D Cs / x') - (c D Cs/2)"
 
Last edited by a moderator:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
You may find that people are going to have a hard time reading the image...can you enlarge it?
 

Caccioppoli

New member
Apr 19, 2013
5
You may find that people are going to have a hard time reading the image...can you enlarge it?
Sorry, I've uploaded a bigger version of the image, split in two figures.
 

Caccioppoli

New member
Apr 19, 2013
5
I can update the problem since I've done some progress.

The following equation

[tex] q=aq^3+b [/tex] [eq#1]

can be approximated with [tex] q=a^{-0.5} + \delta [/tex] [eq#2]

with [tex]\delta=-b/2 [/tex]

Where does this approximation come from and why is [tex]\delta=-b/2[/tex]?

Thank you very much.
 
Last edited:

Caccioppoli

New member
Apr 19, 2013
5
Managed to reach the solution :D

eq.#1 is [tex]aq^3+b=q[/tex]

eq.#1 would be simpler if b=0, the zero-order approximation is

[tex]q_0=aq_0^3[/tex] so [tex]q_0=a^{-0.5}[/tex]

The next order (1st order) approximation is

[tex]q=q_0 + \delta[/tex]

It is assumed that [tex]\delta[/tex] is small in comparison to [tex]q_0[/tex] so that all the terms in [tex]\delta^2[/tex] and [tex]\delta^3[/tex] are discarded and

[tex]q^3=(q_0+\delta)^3≈q_0^3+3q_0^2\delta[/tex]

So that eq.#1 becomes

[tex]q_0+\delta=a[q_0^3+3q_0^2\delta]+b[/tex]

Recalling the zero-order approximation we have that

[tex]\delta=a[3\delta q_0^2]+b[/tex]

then

[tex]\delta=a3\delta a^{-1}+b=3\delta+b[/tex]

The solution is [tex]\delta=-b/2[/tex]
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193
I'm just curious about the context of this problem. It looks like you're doing Laplace Transforms on a PDE (the diffusion equation?). Is that correct?
 

Caccioppoli

New member
Apr 19, 2013
5
The problem is actually of diffusion.

It starts with Fick's First Law of Diffusion, an ODE (which is steady), after it uses a PSEUDO-Steady State Approximation (small t) to get an approximated expression for fluxes.

To understand more about this kind of approximation I should read the main article which is

"Rate of release of medicaments from ointment bases containing drugs in suspension - Higuchi - 1961"


What I've studied till now is a generalization of an expression derived in Higuchi's article of 1961, so I don't know if he starts from a PDE, but probably he does.