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Scaling Invariant, Non-Linear PDE

  1. May 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider the nonlinnear diffusion problem

    [itex] u_t - (u_x)^2 + uu_{xx} = 0, x \in \mathbb{R} , t >0 [/itex]

    with the constraint and boundary conditions

    [itex] \int_{\mathbb{R}} u(x,t)=1, u(\pm \inf, t)=0[/itex]

    Investigate the existence of scaling invariant solutions for the equation and then use the constraint condition to show that

    [itex] u=t^{- \frac{1}{3}}y(z)[/itex] and [itex] z=xt^{- \frac{1}{3}} [/itex]

    2. Relevant equations



    3. The attempt at a solution

    So this is what I did
    Let
    [itex] x = AX [/itex] and [itex] t = BT [/itex] and [itex] u=CU [/itex]

    [itex] \frac{\partial}{\partial x} = \frac{1}{A} \frac{\partial}{\partial X} [/itex]

    [itex] \frac{\partial^2}{\partial x^2} = \frac{1}{A^2} \frac{\partial^2}{\partial X^2} [/itex]

    [itex] \frac{\partial}{\partial t} = \frac{1}{B} \frac{\partial}{\partial T} [/itex]

    so

    [itex] u_t = \frac{1}{B} \frac{\partial u}{\partial T} [/itex]

    [itex] u_x = \frac{1}{A} \frac{\partial u}{\partial X} [/itex]

    [itex] u_{xx} = \frac{1}{A^2} \frac{\partial^2 u}{\partial X^2} [/itex]

    and

    [itex] u_t - (u_x)^2 + uu_{xx} = 0 [/itex] becomes

    [itex] u_T - \frac{B}{A^2} (u_X)^2 - \frac{BC}{A^2}uu_{XX} = 0 [/itex]

    so if [itex] \frac{B}{A^2} = 1[/itex] and [itex] A = \lambda, then B = \lambda^2 [/itex]

    and I want [itex] \frac{BC}{A^2} = 1[/itex] means [itex] C = 1 [/itex]

    so a scaling invariant solution is [itex] u(\lambda x, \lambda^2 t) [/itex]

    but I'm getting that [itex] z = \frac{x}{t^{\frac{1}{2}}} [/itex]

    so I'm not really sure where the

    [itex] z=xt^{- \frac{1}{3}} [/itex]
     
  2. jcsd
  3. May 19, 2013 #2

    haruspex

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    Do you mean [itex] U_T - \frac{B}{A^2} (U_X)^2 - \frac{BC}{A^2}UU_{XX} = 0 [/itex]? If so, I think you dropped a C or two, and reversed a sign.
     
  4. May 19, 2013 #3
    So I redid everything from the point

    [itex] \frac{\partial}{\partial x} u = \frac{C}{A} \frac{\partial U}{\partial X} [/itex]

    [itex] \frac{\partial^2}{\partial x^2} u = \frac{C^2}{A^2} \frac{\partial^2 U}{\partial X^2} [/itex]

    [itex] \frac{\partial}{\partial t} u = \frac{C}{B} \frac{\partial U}{\partial T} [/itex]

    so I get

    [itex] \frac{\partial U}{\partial T} - \frac{CB}{A^2} (\frac{\partial U}{\partial X})^2 - \frac{C^2 B}{A^2}U \frac{\partial^2 U}{\partial X^2} = 0 [/itex]

    so if I let [itex] A=\lambda [/itex] It'll follow that [itex] C=1, B = \lambda^2 [/itex]

    so if I take [itex] \frac{x}{t^{\frac{1}{2}}} = \frac{\lambda X}{(\lambda^2 T)^\frac{1}{2}} = \frac{X}{T^{\frac{1}{2}}} [/itex]

    So essentially I'm not seeing that [itex] z=xt^{\frac{-1}{3}} [/itex]

    I see [itex] z=xt^{\frac{-1}{2}} [/itex]
     
  5. May 20, 2013 #4

    haruspex

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    Sorry, I'm not familiar with this type of problem and I don't know what principles you're applying to find y and z. The OP says to use the constraint conditions for that part, but you don't appear to have done so.
    If you like, I'll post on the Homework Helpers forum to see if someone else can help you.
     
  6. May 20, 2013 #5
    oh yes please do so, that'll be excellent, yes i haven't used the constraint conditions quite yet and to be quite honest I'm not quite sure on how to do so.
     
  7. May 20, 2013 #6

    TSny

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    Shouldn't C just be raised to the first power in the expression above?

    Also, I think the constraint will yield a relation between A and C.
     
  8. May 20, 2013 #7

    haruspex

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    Good catch - I missed that.
     
  9. May 20, 2013 #8
    yup you're right, now lemme see what I can do with this
     
  10. May 20, 2013 #9

    fzero

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    You seem to be on the right track, but it would be simpler to be a little clearer about how the scaling works. I would consider the scaling transformation

    $$ t \rightarrow \lambda t , ~~ x \rightarrow \lambda^a x, ~~ u \rightarrow \lambda^b u.$$

    You can use this to define your ##T,X,U## and it should be a bit clearer since you will get a pair of linear relationships between ##a## and ##b##.
     
  11. May 21, 2013 #10
    hmm, i'm getting this

    [itex] \lambda^{b-1}u_t - \lambda^{2b-2a} (u_x)^2 -\lambda^{b-2a}u_{xx} = 0 [/itex]

    so we'll get a scaling invariant solution if

    [itex] b-1 = 2b-2a = b-2a [/itex] correct?


    I'm getting that [itex] a = - \frac{1}{2}, b = -2 [/itex]

    so if i define [itex] z = \frac{\lambda^{\frac{-1}{2}} x}{(\lambda^{-2}t)^{(\frac{-1}{4})}} [/itex]
    and we get that [itex] z = xt^\frac{-1}{4} [/itex]

    I feel like I should use the constraint based on what TSny said

    I mean if we have that [itex] \int_{\mathbb{R}} u(x,t) dx = 1 [/itex] I feel like it has something to do with

    [itex] \int_{\mathbb{R}} CU(AX,BT) A dX = 1 [/itex]

    any other thoughts?
     
  12. May 21, 2013 #11

    haruspex

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    I get [itex] a = \frac{1}{2}, b = 0 [/itex]
     
  13. May 21, 2013 #12

    fzero

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    You missed a factor of ##u## in the last term.


    Yes, this gives a 2nd linear constraint on the scaling exponents.
     
  14. May 21, 2013 #13

    TSny

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    I could be wrong, but here is how I interpret [itex]u = CU[/itex]:

    ##u(x,t) = CU(X,T)## where ##x = AX## and ##t = CT##.

    Then [itex] \int_{\mathbb{R}} u(x,t) dx =C\int_{\mathbb{R}} U(X,T) dx[/itex]

    Now make the appropriate substitution for ##dx## and see what you get.
     
  15. May 21, 2013 #14
    ALRIGHT,

    I wrestled with this problem but I did it both ways!

    so sticking with it haruspex and TSny I got that

    [itex] \frac{C}{B}U_T - \frac{C^2}{A^2}U_X - \frac{C^2}{A^2}UU_XX = 0 [/itex]

    so if we divide everything by [itex] \frac{C}{B} [/itex] we'll get the equation

    [itex]\frac{CB}{A} = 1[/itex]

    using the constraint

    [itex] \int_{\mathbb{R}} u(x,t)dx = \int_{\mathbb{R}} CU(AX,BT) AdX = 1 \Rightarrow CA = 1 \Rightarrow C = \frac{1}{A}[/itex]

    plugging this back into

    [itex] \frac{CB}{A} = 1 \Rightarrow \frac{B}{A^3} = 1 \Rightarrow B=A^3 [/itex]

    letting [itex] A = \lambda [/itex] means [itex] B = \lambda^3 [/itex]

    so if we let [itex] z = \frac{x}{t^{\frac{1}{3}}} = \frac{X}{T^{\frac{1}{3}}} [/itex]

    similarly doing it your way fzero

    i came up with

    [itex] \lambda^{b-1} = \lambda^{2b-a} [/itex]

    so [itex] b - 2a + 1 = 0 [/itex]

    using the constraint I get that

    [itex] \int_{\mathbb{R}} u(x,t)dx = \int_{\mathbb{R}} \lambda^b u(\lambda^a x, \lambda t) \lambda^a dX = 1 \Rightarrow \lambda^{b+a}= \lambda^0 \Rightarrow b = -a[/itex]

    so

    [itex]-a -2a +1 = 0 \Rightarrow 3a = 1 \Rightarrow a = \frac{1}{3} [/itex] and it follows that [itex]b = - \frac{1}{3} [/itex]

    so using the same idea as above, I get the same result!
     
    Last edited: May 21, 2013
  16. May 21, 2013 #15

    TSny

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    I think you just forgot to type the square on A: [itex]\frac{CB}{A^2} = 1[/itex]

    Otherwise, it all looks good to me.
     
  17. May 21, 2013 #16
    yup it should be A2

    nm i figured that part out and that was garbage.
     
    Last edited: May 21, 2013
  18. May 21, 2013 #17

    fzero

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    You found that the equation is homogenous with respect to the transformation

    $$ t \rightarrow \lambda t , ~~ x \rightarrow \lambda^{1/3} x, ~~ u \rightarrow \lambda^{-1/3} u.$$

    The combination ##z = x/t^{1/3}## is invariant, while ##u## scales like ##t^{-1/3}##. To understand what this means for the form of ##u##, imagine that we knew ##u(t,x)## and that it had a well-defined Taylor series at some point. This Taylor series would have to be of the form

    $$ u(t,x) = \sum_{\alpha,\beta} c_{\alpha\beta} t^\alpha x^\beta.$$

    However, ##u## has a definite scaling property, so each term in this series must also have the same scaling property. Can you work out what sort of constraint is placed on ##\alpha## and ##\beta##? This will lead you to an expression for the Taylor series of ##y(z)## and establish the above relation between ##u## and ##y##.
     
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