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FAS1998
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When using the separation of variable for partial differential equations, we assume the solution takes the form u(x,t) = v(x)*g(t).
What is the justification for this?
What is the justification for this?
I understand that it works in the sense that the solutions it finds are consistent with the differential equations, but how do we know that the solutions couldn’t be more general?Dr Transport said:it works...and it can be shown that the solutions are mathematically rigorous.
FAS1998 said:I understand that it works in the sense that the solutions it finds are consistent with the differential equations, but how do we know that the solutions couldn’t be more general?
Couldn’t solutions of u(x,t) exist that are not of the form v(x)*g(t)?
Or can all functions of u(x,t) be written as v(x)*g(t)?
So the solution of all linear PDE’s take the form u(x,t) = v(x)g(t)?Dr Transport said:For linear equations, you can write them in the form [itex] u(x,t) = v(x)g(t) [/itex]. For non-linear equations that may not be the form of the solution.
No, certainly not.FAS1998 said:So the solution of all linear PDE’s take the form u(x,t) = v(x)g(t)?
Maybe this is contrived, but if u(x,t) is the pdf of a joint distribution where x,t, are not independent, then the equation will not separate.FAS1998 said:So the solution of all linear PDE’s take the form u(x,t) = v(x)g(t)?
Is there a proof or explanation for this somewhere? My textbook doesn’t explain this very thoroughly.
FAS1998 said:So the solution of all linear PDE’s take the form u(x,t) = v(x)g(t)?
Is there a proof or explanation for this somewhere? My textbook doesn’t explain this very thoroughly.
How do you know that there isn’t a more general solution? Couldn’t it be possible that some solution of the form v(x)g(t) exists, but is not the most general form?phyzguy said:No, certainly not.
The idea is that you try to find a solution in this form. If it works, you've found a solution. If it doesn't, you need to try other methods. Usually it doesn't work, but it's worth a try. Here's a similar situation. Say I'm trying to find the square root of 169. It's obviously greater than 10, and less than 20. So I try 11x11=121, 12x12=144, 13x13=169... hey guess what? 13 works! So I've found the solution through a simple method. If I'm trying to find the square root of 170, obviously this method won't work, so i need to try more advanced methods
FAS1998 said:How do you know that there isn’t a more general solution? Couldn’t it be possible that some solution of the form v(x)g(t) exists, but is not the most general form?
Or in other words, how do you know if sepetation or variables has worked, and has provided a general solution?
FAS1998 said:How do you know that there isn’t a more general solution? Couldn’t it be possible that some solution of the form v(x)g(t) exists, but is not the most general form?
Or in other words, how do you know if sepetation or variables has worked, and has provided a general solution?
Separation of variables is a method used to solve partial differential equations (PDEs) by breaking them down into simpler, ordinary differential equations (ODEs) that can be solved algebraically. It involves assuming that the solution to the PDE can be written as a product of functions of each variable, and then using this assumption to find the individual functions.
Separation of variables is applicable for solving PDEs that have boundary conditions that are separable, meaning they can be written as a product of functions of each variable. It is also applicable for PDEs that are linear and homogeneous.
The steps involved in using separation of variables to solve a PDE are:
Separation of variables is a powerful method for solving PDEs because it simplifies the problem by breaking it down into smaller, more manageable ODEs. This allows for the use of well-known techniques for solving ODEs, making the process more straightforward and efficient.
Yes, there are limitations to using separation of variables for solving PDEs. It can only be applied to certain types of PDEs, and even for those, it may not always produce a solution. Additionally, it may not be applicable for PDEs with non-constant coefficients or non-separable boundary conditions.