Change of variables for PDE

  • Thread starter jimbo007
  • Start date
  • #1
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2
hi,
i am having difficulty trying to find a change of variables to solve this partial differential equation
[tex]\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}[/tex]
not sure how to pluck out a change of variables by looking at the equation as its definately not obvious to the untrained eye (me).
have tried
[tex]f(x,t) = p(\eta) [/tex] (not sure if i am even on the right track doing this)
where
[tex]\eta = \frac{x}{t^{\gamma}} [/tex]
but that just gets me in a giant mess with
[tex]-\gamma\frac{x}{t}\frac{\partial p}{\partial \eta} = \frac{\partial ^2 p}{\partial \eta ^2} [/tex]
i may have done [tex]\frac{\partial ^2 f}{\partial x^2}= \frac{\partial \eta}{\partial x}\frac{\partial ^2 p}{\partial \eta ^2}\frac{\partial \eta}{\partial x} [/tex] wrongly.

any suggestions?
thanks for the time
 

Answers and Replies

  • #2
335
4
jimbo007 said:
hi,
i am having difficulty trying to find a change of variables to solve this partial differential equation
[tex]\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}[/tex]
not sure how to pluck out a change of variables by looking at the equation as its definately not obvious to the untrained eye (me).
have tried
[tex]f(x,t) = p(\eta) [/tex] (not sure if i am even on the right track doing this)
where
[tex]\eta = \frac{x}{t^{\gamma}} [/tex]
but that just gets me in a giant mess with
[tex]-\gamma\frac{x}{t}\frac{\partial p}{\partial \eta} = \frac{\partial ^2 p}{\partial \eta ^2} [/tex]
i may have done [tex]\frac{\partial ^2 f}{\partial x^2}= \frac{\partial \eta}{\partial x}\frac{\partial ^2 p}{\partial \eta ^2}\frac{\partial \eta}{\partial x} [/tex] wrongly.

any suggestions?
thanks for the time

I don't know about a substitution, but if you're just after a solution you can set f(x,t) = X(x)T(t) and the equation becomes separable.

-Dan
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,833
964
There is no need to get "x" involved in your change of variable. You want to go from
[tex]\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}[/tex]
or, similarly,
[tex]t^{-\gamma}\frac{\partial f}{\partial t} = \frac{\partial ^2 f}{\partial x^2}[/tex]
to something like
[tex]\frac{\partial f}{\partial \eta} = \frac{\partial ^2 f}{\partial x^2}[/tex]
which means you want
[tex]\frac{\partial f}{\partial \eta}= \frac{\partial f}{\partial t}\frac{dt}{d \eta}= t^{-\gamma}\frac{\partial f}{\partial t}[/tex]
So you must have
[tex]\frac{dt}{d\eta}= t^{-\gamma}[/tex]
or
[tex]\frac{d\eta}{dt}= t^{\gamma}[/tex]
 

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