PDE and the separation of variables

In summary, the conversation discusses using the equation ##u(x,y)=f(x)g(y)## to solve a given partial differential equation. The speaker initially substitutes the values of ##u_{xx}## and ##u_{yy}## in the PDE and solves the resulting ODEs, but struggles with understanding ##u_{t}##. They suggest setting ##u_{t}=0## due to the function only depending on x and y, but question if this is the correct approach. It is then mentioned that the question may have been written in a confusing manner, and that the solution may involve making the leap to u = h(t)f(x,y). The conversation ends with the question of how to solve the equation.
  • #1
Magnetons
18
4
Homework Statement
Apply the method of separation of variables ##u(x,y)=f(x)g(y)## to solve the equation .
Relevant Equations
##u_{t}=c^{2}(u_{xx}+u_{yy})##
using the equation ##u(x,y)=f(x)g(y)##, first, I substitute the value of ##u_{xx}## and ##u_{yy}## in the given PDE. after that solve the ODEs but I can't understand about the ##u_{t}##.In my solution, I put ##u_{t}=0## because u is only the function of x and y. Is it the right approach, to me it seems wrong
 

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  • #2
Aren't you supposed to separate in a different way ? In u(t) and u(x,y) for example :wink:

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  • #3
Google 'heat equation'
 
  • #4
BvU said:
Aren't you supposed to separate in a different way ? In u(t) and u(x,y) for example :wink:

##\ ##

don't know it is how the question is given in the book
 
  • #5
Well, you run into trouble with ##u_t=0##, so I suggest to try something different.

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  • #6
BvU said:
Well, you run into trouble with ##u_t=0##, so I suggest to try something different.

##\ ##
something different ..
 
  • #7
Magnetons said:
Homework Statement: Apply the method of separation of variables ##u(x,y)=f(x)g(y)## to solve the equation .
Relevant Equations: ##u_{t}=c^{2}(u_{xx}+u_{yy})##

using the equation ##u(x,y)=f(x)g(y)##, first, I substitute the value of ##u_{xx}## and ##u_{yy}## in the given PDE. after that solve the ODEs but I can't understand about the ##u_{t}##.In my solution, I put ##u_{t}=0## because u is only the function of x and y. Is it the right approach, to me it seems wrong

If they wanted you to assume [itex]u_t = 0[/itex], would they not have just asked for [itex]u_{xx} + u_{yy} = 0[/itex]?

Perhaps you are expected to make the leap to [itex]u = h(t)f(x,y)[/itex].
 
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  • #8
pasmith said:
If they wanted you to assume [itex]u_t = 0[/itex], would they not have just asked for [itex]u_{xx} + u_{yy} = 0[/itex]?

Perhaps you are expected to make the leap to [itex]u = h(t)f(x,y)[/itex].
No ## u_t = 0 ## doesn't mention in question i assume it .
 

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  • #9
Magnetons said:
don't know it is how the question is given in the book
Yeah, they managed to confuse you (on purpose?) writing ##u(x,y)=f(x)g(y)## instead of ##u(p,q)=f(p)g(q)## or something less suggestive...

Your post #8 explains why. (and post#7 IS exercise 25 (h) ! )

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  • #10
BvU said:
Yeah, they managed to confuse you (on purpose?) writing ##u(x,y)=f(x)g(y)## instead of ##u(p,q)=f(p)g(q)## or something less suggestive...

Your post #8 explains why. (and post#7 IS exercise 25 (h) ! )

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how should I solve this equation
 
  • #11
pasmith said:
make the leap to [itex]u = h(t)f(x,y)[/itex].
 
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  • #12
Found a satisfactory solution ?

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1. What is PDE and how is it different from ordinary differential equations (ODEs)?

PDE stands for partial differential equations, which are equations that involve multiple variables and their partial derivatives. This is different from ODEs, which only involve one independent variable and its derivatives.

2. How is the separation of variables method used to solve PDEs?

The separation of variables method involves assuming a solution to the PDE that can be written as a product of functions of each variable. This allows the PDE to be separated into individual ODEs, which can then be solved using standard techniques.

3. Can the separation of variables method be used for all types of PDEs?

No, the separation of variables method is only applicable to certain types of PDEs, such as linear and homogeneous equations with boundary conditions that can be separated. It is not effective for nonlinear or non-homogeneous PDEs.

4. Are there any limitations to using the separation of variables method?

Yes, the separation of variables method can only be used for PDEs with certain types of boundary conditions, such as Dirichlet or Neumann boundary conditions. It also may not always yield a complete solution to the PDE.

5. Are there any real-world applications of PDE and the separation of variables method?

Yes, PDEs and the separation of variables method are used in many fields of science and engineering, such as physics, chemistry, and engineering. They are commonly used to model and solve problems related to heat transfer, fluid dynamics, and wave phenomena.

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