Diffusion Equation (Coordinate dependent diffusion coeff)

In summary, the conversation discusses the difficulty of finding the Green's function for a diffusion equation with a coordinate-dependent diffusion coefficient. The person has tried two methods, separation of variables and Fourier transforms, but neither were successful. They are seeking help and mention that the Green's function can be derived using a Fourier transform technique, with more details in the book Partial Differential Equations - An Introduction by Walter A. Strauss. The Green's function for this equation is given by a formula involving the diffusion coefficient and time variables.
  • #1
andreasgeo
4
0
Hello... My main problem is to find the Green's function of this pde:
[itex]\frac{∂T(x,t)}{∂t}[/itex]=[itex]\frac{∂}{∂x}[/itex](D(x)[itex]\frac{∂T(x,t)}{∂x}[/itex]) .

This is the well-known diffusion equation, but here, the diffusion coefficient is coordinate-dependent.
My main focus is to find the Green's function, but i know that in order to find it, i have to solve this equation... I have tried 2 methods, the separation of variables and The Fourier transforms.. Both didnt help...
Any idea? Thanks a lot in advance...

PS: Ofc T(x,t) is tempereture
 
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  • #2
, and D(x) is the diffusion coefficient.The Green's function for this equation is given by: G(x,t;x',t') = \frac{1}{2 \sqrt{\pi D(x) (t-t')}} \exp\left[-\frac{(x-x')^2}{4 D(x) (t-t')}\right] This expression can be derived using an appropriate Fourier transform technique. The details of the derivation can be found in the book Partial Differential Equations - An Introduction by Walter A. Strauss.
 

1. What is the diffusion equation?

The diffusion equation is a partial differential equation that describes the time evolution of a quantity undergoing diffusion, such as heat or particles, in a given space. It is represented by the equation ∂C/∂t = D∇2C, where C is the concentration or density of the diffusing substance, t is time, and D is the diffusion coefficient.

2. What is the role of the coordinate dependent diffusion coefficient in the diffusion equation?

The coordinate dependent diffusion coefficient, often denoted as D(x,y,z), is a function of the spatial coordinates that accounts for the heterogeneity of the diffusing medium. It takes into consideration the varying properties, such as temperature or density, that can affect the diffusion process in different regions of the space.

3. How does the diffusion coefficient affect the rate of diffusion?

The diffusion coefficient is directly proportional to the rate of diffusion, meaning that a higher diffusion coefficient leads to a faster diffusion rate. This is because a larger diffusion coefficient indicates a higher mobility of the diffusing substance, allowing it to spread more quickly through the medium.

4. Can the diffusion equation with coordinate dependent diffusion coefficient be solved analytically?

In most cases, the diffusion equation with a coordinate dependent diffusion coefficient cannot be solved analytically. This is because the diffusion coefficient D(x,y,z) is often a complex function of the coordinates and cannot be easily integrated. Numerical methods, such as finite difference or finite element methods, are commonly used to approximate solutions to this type of diffusion equation.

5. What are some practical applications of the diffusion equation with coordinate dependent diffusion coefficient?

The diffusion equation with coordinate dependent diffusion coefficient has many applications in various fields, such as physics, chemistry, biology, and engineering. Some examples include modeling heat transfer in materials with varying thermal conductivity, studying the spread of contaminants in environmental systems, and simulating drug diffusion in biological tissues.

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