How to separate variables in this PDE?

In summary, the conversation involves a person trying to solve a PDE with known functions. They are exploring different methods, such as separating F(x,t) and using a change of variable, but have not found a solution yet. They are also discussing notations and ways to solve the PDE analytically.
  • #1
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TL;DR Summary
Can anyone suggest a way to separate the variables in this PDE, so I can solve it analytically?
My PDE:
F,x,t + A(x)*F(x,t)*[(x+t)^(-3/2)] = 0

A(x) is a known function of x.
Trying to separate F(x,t) like
F(x,t) = F1(x)*F2(t)*F3(x+t).

I’m getting desperate to solve,
any suggestions??
 
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  • #2
Make the change of variable [itex](\zeta, \eta) = (x, x + t)[/itex]. Then the Ansatz [itex]F = H(\eta)Z(\zeta)[/itex] yields [tex]
H'(\eta)Z'(\zeta) + H''(\eta)Z(\zeta) + A(\zeta)H(\eta)Z(\zeta)\eta^{-3/2} = 0.[/tex] I'm not sure this is separable.
 
  • #3
Thanks pasmith,
I did try something like that, and basically got that same type of equation, but I’m not sure what to do with it next. Still searching…
 
  • #4
It wwould surely be easier to parse in ##LaTeX##
 
  • #5
[tex]F_{,x,t} + \frac{A(x)}{(x+t)^{3/2}} F(x,t) = 0[/tex]

[itex]A(x)[/itex] is messy but known; [itex]F(x,t)[/itex] must be solved for analytically, through separability or any other way.
 
Last edited:
  • #6
What is ##F_{,x,t} ##?
 
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  • #7
Common notation, [itex]F_{,x,t} \equiv \frac{\partial ^{2} F(x,t)}{\partial x \partial t}[/itex]
 
  • #8
Wow I've never seen it with the commas. OK thanks.
 
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1. What is a partial differential equation (PDE)?

A partial differential equation (PDE) is a mathematical equation that involves partial derivatives of a dependent variable with respect to multiple independent variables. It is commonly used to model physical phenomena in fields such as physics, engineering, and economics.

2. How do you identify the variables in a PDE?

The variables in a PDE can be identified by looking at the terms in the equation. The dependent variable is the one that is being differentiated, while the independent variables are the ones that the dependent variable is being differentiated with respect to.

3. What is the process for separating variables in a PDE?

The process for separating variables in a PDE involves isolating the dependent variable on one side of the equation and the independent variables on the other side. This is done by using algebraic manipulation and applying boundary conditions to solve for the constants of integration.

4. Are there any specific techniques for separating variables in different types of PDEs?

Yes, there are various techniques for separating variables in different types of PDEs. Some common techniques include the method of characteristics, the method of eigenfunction expansion, and the method of separation of variables. The specific technique used will depend on the type and complexity of the PDE.

5. Can all PDEs be solved by separating variables?

No, not all PDEs can be solved by separating variables. This method is only applicable to certain types of linear PDEs with specific boundary conditions. For more complex PDEs, other techniques such as numerical methods or series solutions may be necessary.

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