Solving Fick's Law by Finite Difference Method

In summary, the conversation discusses the use of Finite Difference Method to solve a Fick's Law equation, instead of using Separation of Variables as previously attempted. The person is seeking guidance on how to use this method for their MatLab simulation project. They mention difficulties in understanding the concept and are seeking help from others.
  • #1
Stefanie
4
0
Hi, i need help in solving a Fick's Law [ (∂c_k)/∂t = D_k (∂^2 c_k)/(∂x^2 ) ] by Finite Difference Method.
Previously, I tried solving the Fick's Law by using the Separation of Variable method but that was not the correct way as told by my Prof as the correct way is to use Finite Difference Method.

Can anyone teach or guide me on how to solve as I had never learn finite difference method before?

Thanks alot.
 
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  • #2
What, exactly, is the problem you are trying to solve? The "finite difference method" is a method of numerically solving differential equations where we are given values at the boundary of a region. Whether "separation of variables" will work on any partial differential equation depends strongly on the geometry of the region. "Fick's law" (also called the "diffusion" or "heat" equation) certainly can be solved by separation of variables on a rectangular region (Fourier series solutions), on a circular or cylindrical region (series of Bessel functions) or on a spherical region (series of spherical harmonics).
 
  • #3
My project is actually on a MatLab simulation but since I'm not required to write out the source code, my Prof wants me to show how can this differential equation can be used to solve using finite difference method instead of a separation variable method. For instance, to transfer the differential equation into the same form as finite difference method.

I look up a few reference book but i still can't seems to understand. Hence, will like to seek help from here.

Thanks.
 

What is Fick's Law?

Fick's Law is a mathematical equation that describes the diffusion of a substance through a medium. It states that the rate of diffusion is proportional to the concentration gradient of the substance and the diffusion coefficient of the medium.

What is the Finite Difference Method?

The Finite Difference Method is a numerical technique used to solve differential equations by approximating the derivatives of a function using a finite set of data points. It is commonly used in scientific and engineering applications where analytical solutions are not feasible.

Why is Fick's Law solved using the Finite Difference Method?

Fick's Law is a second-order differential equation, which means that it requires two boundary conditions to solve. The Finite Difference Method is well-suited for solving this type of equation as it allows for the approximation of derivatives at the boundary points, making it easier to obtain a numerical solution.

What are the steps involved in solving Fick's Law by Finite Difference Method?

The steps involved in solving Fick's Law by Finite Difference Method include: defining the boundary conditions, discretizing the equation into a finite number of points, constructing a difference equation using the discretized points, solving the difference equation for each point, and finally, evaluating the solution at the desired points.

What are the advantages of using the Finite Difference Method to solve Fick's Law?

The Finite Difference Method offers several advantages when solving Fick's Law, including its ability to handle complex boundary conditions, its flexibility in solving higher-order differential equations, and its efficiency in handling multidimensional problems. Additionally, it provides a more accurate solution compared to analytical methods when dealing with non-linear or time-dependent systems.

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