1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Let v(x,t) = u(x+ct) and show that

  1. Mar 19, 2012 #1
    1. The problem statement, all variables and given/known data

    NOTE: I realize the following is a partial differential equation, but I believe the answer to my question is a straight forward multi-variate calculus question.

    Let v(x,t) = u(x+ct,t) and show that v(x,t) solves the following...

    ut = κ uxx ; (-∞<x<∞)

    u(x,0) = [itex]\phi[/itex](x)

    NOTE: You may disregard the following, if necessary for you to answer this question:

    u(x,0) = [itex]\phi[/itex](x)

    and simply show that v(x,t) solves the following...

    ut = κ uxx

    ALTERNATIVELY, you may help me by commenting on the correctness of my work below.




    2. Relevant equations

    See above and below...




    3. The attempt at a solution

    v(x,t) = u(x+ct,t)

    v'(x,t) = c u[itex]_{x}[/itex](x+ct,t) + u[itex]_{t}[/itex](x+ct,t)

    v[itex]_{x}[/itex](x,t) = c u[itex]_{x}[/itex](x+ct,t)

    v[itex]_{xx}[/itex](x,t) = c[itex]^{2}[/itex] u[itex]_{xx}[/itex](x+ct,t)

    v[itex]_{t}[/itex](x,t) = u[itex]_{t}[/itex](x+ct,t)
     
  2. jcsd
  3. Mar 19, 2012 #2
    Any help on any part will be GREATLY appreciated!!!
     
  4. Mar 19, 2012 #3

    cepheid

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Welcome to PF,


    Just before we start, are you sure it was not meant to be simply v(x,t) = u(x+ct)? Because I know that a function in which the temporal and spatial dependence are related in this way will satisfy a wave equation, which is given by utt = κuxx. This is different from the PDE that you gave. But anyway, taking what you wrote at face value:


    First of all, I don't think that v' (v "prime") is meaningful here. In single-variable calculus, a prime symbol typically means "derivative with respect to the argument." But this function has two arguments. Since it is a multi-variable function, any derivative is going to be a partial derivative, and you must explicitly specify which variable you are differentiating with respect to. As for your attempt at partial derivatives, my approach would be to write:

    u = u(w,t) where w = x+ct

    then,

    [tex]\frac{\partial u}{\partial x} = \frac{\partial u}{\partial w}\frac{\partial w}{\partial x}[/tex][tex]= \frac{\partial u}{\partial w}[/tex]

    Et cetera for the other derivatives
     
  5. Mar 19, 2012 #4
    To your first point...

    I am sure it is:

    Let v(x,t) = u(x+ct,t)

    to your second point...

    That would mean the following is incorrect, right?

    v.sub.x(x,t) = c u.sub.x(x+ct,t)

    and should have been...

    v.sub.x(x,t) = u.sub.x(x+ct,t)
     
  6. Mar 19, 2012 #5

    cepheid

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What I was saying was that v_x = u_w, with underscores denoting subscripts.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook