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?it works.....and it can be shown that the solutions are mathematically rigorous.
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.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)?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.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.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.
One you have that form u(x,t) = v(x)g(t) and you found the forms of v(x) and g(t) from their ODEs (and possibly some of the boundary and initial conditions),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?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
For most PDEs where you are applying the method of separation of variables, there is a proof of uniqueness. So it you have found a solution, you have found the solution.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?
Uniqueness theorems: If you find a solution, by whatever ad hoc non-a-priori-justified method, then it must be the solution.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?