Simple Substituting and Rearranging

Caccioppoli

New member

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

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

Caccioppoli

New member
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
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.

Last edited:

Caccioppoli

New member
Managed to reach the solution

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

Indicium Physicus
Staff member
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
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.