Can S2 Solve the Diffusion Equation in 2D?

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Homework Help Overview

The discussion revolves around solving the diffusion equation in two dimensions, specifically examining the function S2 = S(x,t)S(y,t) and its relation to the equation St = k (Sxx + Syy).

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the relationship between S(x,t) and S(y,t) and their potential solutions to the one-dimensional diffusion equation. Questions arise regarding the assumptions that can be made about these functions.

Discussion Status

There is an ongoing exploration of the implications of assuming S(x,t) and S(y,t) are solutions to the one-dimensional diffusion equation. Some participants suggest dividing through by S(x,t)S(y,t) to derive further insights, indicating a productive direction in the discussion.

Contextual Notes

Participants note the lack of explicit assumptions provided in the problem regarding the nature of S(x,t) and S(y,t), which may affect the approach to solving the equation.

theneedtoknow
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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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Presumably S(x,t) is the solution to a closely related equation?
 
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?
 
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.
 

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