Scaling Problem in Diffusion Equation

In summary, the conversation discusses the diffusion equation and its transformation into an ODE through scaling. It is mentioned that the function f(η) is not invariant with respect to x and t, as seen from the non-zero partial derivatives of f with respect to x and t. This means that f(η) can provide physical information about the system, taking into account the scaling parameter η.
  • #1
kayahan
1
0
Hello everyone,

I have a question that is bothering me a bit. I would be happy if you could give an idea or tell me a specific point to look at.

Lets say that we have an arbitrary function that obeys diffusion equation:

f = f(η), here η is the scaling parameter for pde which equals to x/√t
(Diffusion eq. becomes [itex]df/d\eta=-2\eta d^{2}f/d\eta^{2}[/itex] after scaling)

As I understood, diffusion eq. becomes an ode after scaling and it contains all the possible solutions for x and t pairs in η. Can we say that f(η) is invariant with respect to x and t? or in other words say ∂f/∂t=0 or not? What kind of physical information can we get just by lloking at f(η)?

Thanks in advance!
 
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  • #2
No, [itex]\partial f/\partial x[/itex] and [itex]\partial f/\partial x[/itex] are NOT 0 because [itex]\eta[/itex] is itself a function of x and t.

[tex]\frac{\partial f}{\partial x}= \frac{df}{d\eta}\frac{\partial \eta}{\partial x}= \frac{1}{\sqrt{t}}\frac{df}{d\eta}[/tex]
[tex]\frac{\partial f}{\partial t}= \frac{df}{d\eta}\frac{\partial \eta}{\partial t}= -\frac{x}{\sqrt{t^3}}\frac{df}{d\eta}[/tex]
 

What is the scaling problem in diffusion equation?

The scaling problem in diffusion equation refers to the difficulty in accurately predicting the behavior of a system when the length and time scales of the system are changed. This is due to the fact that diffusion equations are scale-dependent and small changes in the length and time scales can result in significant changes in the behavior of the system.

Why is the scaling problem important in scientific research?

The scaling problem is important in scientific research because it can affect the accuracy and reliability of predictions and experiments. If the scaling problem is not taken into account, the results may not be applicable to different length and time scales, leading to erroneous conclusions.

What are some common techniques used to address the scaling problem?

Some common techniques used to address the scaling problem include non-dimensionalization, which involves normalizing the variables and parameters in the diffusion equation, and asymptotic analysis, which involves studying the behavior of the system as the length and time scales approach zero or infinity.

How does the scaling problem affect diffusion in real-world systems?

The scaling problem can affect diffusion in real-world systems by altering the diffusion rates and patterns. For example, in biological systems, changes in length and time scales can affect the diffusion of molecules and nutrients, which can have an impact on cellular processes and functions.

Are there any current solutions to the scaling problem?

While there is no definitive solution to the scaling problem, ongoing research and advancements in computational methods have made it possible to better understand and account for the scaling effects in diffusion equations. Additionally, interdisciplinary collaborations and the use of multiple techniques can also help mitigate the scaling problem in scientific research.

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