Can S2 Solve the Diffusion Equation in 2D?

In summary, the equation St = k (Sxx + Syy) can be solved using the solution S(x,t)S(y,t), which can be found by dividing through by S(x,t)S(y,t) and solving for S(x,t). This equation is not exactly the 1d diffusion equation, but it is simple to solve.
  • #1
theneedtoknow
176
0
Show that S2 = S(x,t)S(y,t) solves St = k (Sxx + Syy)

well
St = St(x,t)S(y,t) + S(x,t)St(y,t)
Sxx = Sxx(x,t)S(y,t)
Syy=Syy(y,t)S(x,t)

but what do i do from there?
 
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  • #2
Presumably S(x,t) is the solution to a closely related equation?
 
  • #3
Well yeah, I thought about assuming that S(x,t) and S(y,t) solve the 1d diffusion equation in their respective dimensions, and then it's easy to just replace all the Sts with the Sxx and Syys, but the question doesn't provide any assumptions about S(x,t) and S(y,t) being solutions to the 1d equation. Is that just something I should assume anyway?
 
  • #4
If you divide through by S(x,t)S(y,t) you will be able to find the equation that S(x,t) must satisfy. It's not quite the 1d diffusion equation, but it looks easy to solve.
 

1. What is the diffusion equation in 2D?

The diffusion equation in 2D is a partial differential equation that models the spread of a substance over a two-dimensional space due to random movement of its particles. It is also known as the heat equation or the diffusion equation in physics and chemistry.

2. What does the diffusion coefficient represent in the diffusion equation?

The diffusion coefficient, represented by the symbol D, is a constant that determines the rate at which the substance diffuses through the medium. It is dependent on factors such as the temperature, viscosity, and size of the particles.

3. How is the diffusion equation solved in 2D?

The diffusion equation in 2D can be solved using mathematical techniques such as separation of variables, Fourier series, or numerical methods such as finite difference or finite element methods. The specific method used depends on the boundary conditions and initial conditions of the problem.

4. What are the applications of the diffusion equation in 2D?

The diffusion equation in 2D has various applications in fields such as physics, chemistry, biology, and engineering. It is used to model diffusion of heat, mass, and particles in different systems such as heat transfer in buildings, drug delivery in biological systems, and diffusion of pollutants in the environment.

5. What are the limitations of the diffusion equation in 2D?

The diffusion equation in 2D assumes that the diffusion process is homogeneous, isotropic, and follows Fick's law. This may not be the case in some real-world scenarios, leading to limitations in its accuracy. Additionally, it does not take into account convection or advection, which can significantly affect the diffusion process in some situations.

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