Understanding the Separation of Solutions in Partial Differential Equations"

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In summary, when solving a partial differential equation, you can not generally solve it by dividing with F(r,t). Instead, you solve it by using the continuity of F(r,t).
  • #1
ChrisVer
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I am not sure again, whether this belongs here or in mathematics.
When we have a partial differential equation, in general we can write the solution of [itex]F(r,t)[/itex] as:
[itex]F(r,t)=R(r)T(t)[/itex]
In the procedure of separating the differential equation, we find ourselves dividing with [itex]F(r,t)[/itex].
Isn't that actually problematic for points where [itex]R[/itex] or [itex]T[/itex] happen to be zero? I know that they can't be zero everywhere coz the solution would be trivial, but what stops them from being zero at distinct points? Actually nothing...
The only solution to this problem would be the imposing of continuity at the point of interest, setting the left side solution equal to the right side. But dividing with zero is still a problem :(

Thanks.
 
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  • #2
ChrisVer said:
I am not sure again, whether this belongs here or in mathematics.
When we have a partial differential equation, in general we can write the solution of [itex]F(r,t)[/itex] as:
[itex]F(r,t)=R(r)T(t)[/itex]
Actually, you can not do this in general.

Also this is not something particular to quantum mechanics.
 
  • #3
MisterX said:
Actually, you can not do this in general.

Also this is not something particular to quantum mechanics.

when can you do it?

I know, I was skeptic about it in the first place... I just came across it right now in QM for Schrod equation solving, and the question just stroke in my head.
For example, in the Hydrogen atom, there exist [itex]r[/itex] such that [itex]F,R(r)=0[/itex]
 
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  • #4
Your question is more easily answered by a physicist than by a mathematician.
Physicists answer: Although you do write F/(RT), tha this only temporary.
You never actually divide by the R unless it is in the form R(r)/R(r)=1.
I something like R"/R=k, the first thing you do is change it to R"=kR.
 
  • #5
ChrisVer said:
when can you do it?

It's quite tricky to find a simple and general criterion, but I remember seeing it be done for classes of equations, e.g. diffusion equations. If you are looking for a general answer, it probably will have something to do with symmetries and be rather distanced from how physicists would tend to proceed.

In practice, you do what one of my differential equations professors used to tell us: try the separation of variables method to every equation. If it works, it works and you probably saved a lot of time. If it doesn't work, this becomes clear after two steps of calculation or so, so not much time was lost there.
 
  • #6
hmm I think it's OK... what is your opinion about the continuity criterion?
I mean, what about if we indeed divide by [itex]F(r,t)[/itex] without caring about the zeros of it? but instead, solve automatically for those points by using the continuity of the [itex]F(r,t)[/itex] function?
Let's say that [itex]R(r_{0})=0[/itex] for some radius [itex]r_{0}[/itex]... then we can solve it at [itex]R(r_{0}\pmε)[/itex] and then send [itex]ε \rightarrow 0[/itex] and imposing:
[itex]R(r_{0}+ε)=R(r_{0}-ε)[/itex]

I think mathematically that's not incorrect (?)
 
  • #7
ChrisVer said:
hmm I think it's OK... what is your opinion about the continuity criterion?
I mean, what about if we indeed divide by [itex]F(r,t)[/itex] without caring about the zeros of it? but instead, solve automatically for those points by using the continuity of the [itex]F(r,t)[/itex] function?
Let's say that [itex]R(r_{0})=0[/itex] for some radius [itex]r_{0}[/itex]... then we can solve it at [itex]R(r_{0}\pmε)[/itex] and then send [itex]ε \rightarrow 0[/itex] and imposing:
[itex]R(r_{0}+ε)=R(r_{0}-ε)[/itex]

I think mathematically that's not incorrect (?)

Generally, one would treat the case of [itex]R=0[/itex] separetely, and solve the differential equation for this case too, which is obviously permitted. Your method is essentially the same, if one matches the solutions using the smoothness property (the derivative of the wave function is continuous too), just as in a potential barrier problem. So yeah, I think it should work fine for the Schrödinger equation.

However, it can't be a general method for all partial differential equations. Non-linear PDEs (for which separation of variables admittedly don't work as often) and linear PDE problems with non-smooth initial conditions can give rise to discontinuous solutions, for example.
 

FAQ: Understanding the Separation of Solutions in Partial Differential Equations"

1. What is the concept of "separation of solutions" in partial differential equations?

The concept of "separation of solutions" in partial differential equations refers to a technique used to solve these types of equations by breaking down the solution into simpler parts that can be solved individually. This involves finding a set of functions that, when multiplied together, form a solution to the original equation.

2. Why is "separation of solutions" important in understanding partial differential equations?

"Separation of solutions" is important in understanding partial differential equations because it allows for the solution to be expressed in terms of simpler functions, making it easier to analyze and solve. It also helps to reduce the complexity of the equation and make it more manageable for further calculations.

3. Can any partial differential equation be solved using the "separation of solutions" method?

No, not all partial differential equations can be solved using the "separation of solutions" method. This technique is most commonly used for linear partial differential equations with constant coefficients. Nonlinear or variable coefficient equations may require different methods for solving.

4. What are the steps involved in using the "separation of solutions" method to solve a partial differential equation?

The steps involved in using the "separation of solutions" method are as follows:

  1. Assume a solution in the form of a product of functions.
  2. Substitute the assumed solution into the original partial differential equation.
  3. Separate the resulting equation into individual terms, each containing only one of the assumed functions.
  4. Set each term equal to a constant and solve for the corresponding function.
  5. Combine the individual solutions to get the final solution to the partial differential equation.

5. Are there any limitations to using the "separation of solutions" method for solving partial differential equations?

Yes, there are some limitations to using the "separation of solutions" method. This technique may not be applicable for all types of partial differential equations, and the resulting solution may not always be valid for the entire domain of the equation. Additionally, the process of separating the equation into individual terms and solving for each function can be time-consuming and may not always lead to a closed-form solution.

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