Simple Substituting and Rearranging

[Caccioppoli]

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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)"

 

[MarkFL]

You may find that people are going to have a hard time reading the image...can you enlarge it?

 

[Caccioppoli]

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]

I can update the problem since I've done some progress.

The following equation

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

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

with $$\delta=-b/2$$

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

Thank you very much.

 

[Caccioppoli]

Managed to reach the solution :D

eq.#1 is $$aq^3+b=q$$

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

$$q_0=aq_0^3$$ so $$q_0=a^{-0.5}$$

The next order (1st order) approximation is

$$q=q_0 + \delta$$

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

$$q^3=(q_0+\delta)^3≈q_0^3+3q_0^2\delta$$

So that eq.#1 becomes

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

Recalling the zero-order approximation we have that

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

then

$$\delta=a3\delta a^{-1}+b=3\delta+b$$

The solution is $$\delta=-b/2$$

 

[Ackbach]

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]

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.