Solution to diffusion equation - different input

[Dave007]

Hi,

I have seen the solution to the diffusion equation written as C=(N/sqrt(4PiDt))exp(-x^2/4Dt). Hoever, as I understand it, this is for an instant input of N material. I want to express the concentration of substance at a point x away from the source for an arbitrary input signal. Is there any nice way to do this please?

Thanks,
Dave

 

[jambaugh]

The solution you give can be translated by replacing x with x-a to give a distribution centered around x=a. You can also superpose many solutions since the diffusion equation is linear.

 

[Dave007]

Thanks jambaugh.

I've actually modified the equation to include an advective part by making x = (x+vt). However, what I want to do is change N to be a function of time. This may require a different equation because as soon as I do that, it doesn't satisfy the diffusion equation anymore I don't think. But I am a bit confused by it all.

Essentially what I want is this:
I have an ion transient through channels in a cell and I want to represent that transient at a point 'x' away from those channels. I thought the diffusion (advective-diffusion) equation is perfect. However, the solution I found is only valid for an initial injection of substance. I want to continually be injecting substance at a varying rate. Is there any solution you know of that would enable me to do this?

Thanks!

 

[jambaugh]

Thanks jambaugh.

I've actually modified the equation to include an advective part by making x = (x+vt). However, what I want to do is change N to be a function of time. This may require a different equation because as soon as I do that, it doesn't satisfy the diffusion equation anymore I don't think. But I am a bit confused by it all.

Essentially what I want is this:
I have an ion transient through channels in a cell and I want to represent that transient at a point 'x' away from those channels. I thought the diffusion (advective-diffusion) equation is perfect. However, the solution I found is only valid for an initial injection of substance. I want to continually be injecting substance at a varying rate. Is there any solution you know of that would enable me to do this?

Thanks!

Introducing (uniform constant velocity) advection should be equivalent to choosing a moving coordinate system. This will alter the differential equation but it is an equivalent problem in that the solutions to the regular diffusion equation, once the velocity transform is applied, will be solutions to the advective diffusion equation.

If your velocity is a function of position and/or time then things are going to get nasty and I'm not sure there are simple methods. You may need to execute a Finite Elements Model to numerically solve the equation. Look around the web for numerical packages which may work.

As far as continuously adding substance you are now talking about an inhomogeneous diffusion equation:
du/dt = D d^2u/dx^2 + f(x,t)


where u(x,t) be the concentration of substance at a given time and position and f is the source term.

There's much literature on solving the diffusion (heat) equation and a great deal of it is online. Look into the Green's function approach and/or solutions via Fourier transforms.

 

[Dave007]

As far as I can see, though, the solutions all depend on knowing what f(x,t) is. In my case, I can't express it as a function. Is there any easy way to solve this?

Thanks for all your help!

 

[Borek]

You may want to consult electrochemist working with voltammetric methods. They deal with similar problems all the time. IMHO simple answer to the question

Is there any easy way to solve this?

is NO.

 

[Dave007]

Any electrochemist who works with voltammetric methods in the house?

PLEASE?

 

[matematikawan]

Thanks jambaugh.

... I thought the diffusion (advective-diffusion) equation is perfect. However, the solution I found is only valid for an initial injection of substance. I want to continually be injecting substance at a varying rate. Is there any solution you know of that would enable me to do this?

Thanks!

Postulates of the heat/diffusion equation are only approximations to the physical situation. But they are good approximations. This is what I have recently learned.

https://www.physicsforums.com/showthread.php?t=308414

 

[matematikawan]

I do not think it is necessary to believe that the same God who has given us our senses, reason, and intelligence wished us to abandon their use, giving us by some other means the information that we could gain through them. -Galileo Galilei

Why should he abandon. Only through reason and observation that we know he exist!

 

[zinovys]

Would you explain how can you present f(x,t)?

As far as I can see, though, the solutions all depend on knowing what f(x,t) is. In my case, I can't express it as a function. Is there any easy way to solve this?

Thanks for all your help!