Results 1 to 9 of 9

Math Help - Solving a PDE ???

  1. #1
    Junior Member
    Joined
    Apr 2009
    Posts
    70

    Solving a PDE ???

    Hey all,

    I am trying to determine how many solns. the PDE Ut = Uxx has. However, I have no idea how to solve this PDE.

    Unlike ODE's which you are given a fairly "standard" y and x variable equation, PDE's just lose me from the beginning. Please show me how to solve such an equation as above, because I have about four others to solve and have no idea how to determine the number of solns to each. I am sure that if I am shown once, I can get the rest.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,367
    Thanks
    42
    Quote Originally Posted by spearfish View Post
    Hey all,

    I am trying to determine how many solns. the PDE Ut = Uxx has. However, I have no idea how to solve this PDE.

    Unlike ODE's which you are given a fairly "standard" y and x variable equation, PDE's just lose me from the beginning. Please show me how to solve such an equation as above, because I have about four others to solve and have no idea how to determine the number of solns to each. I am sure that if I am shown once, I can get the rest.

    Thanks.
    Typically with the PDE u_t = u_{xx} you're given boundary conditions and initial conditions. Do you have these?

    On a side note - unlike ODEs, PDEs (without BC's and IC's) usually have an infinite number of solutions. Ex.

    y'' + y = 0 solutions y = \sin x, y = \cos x

    u_t = u_{xx} has, as one class of solution

    u = e^{-k^2 t} \sin k x , u = e^{-k^2 t} \cos k x for any number k.

    It also has polynomial solutions

     <br />
u = x^2 + 2t<br />
     <br />
u = x^3 +6 x t<br />
     <br />
u = x^4 + 12 x^2 t + 12 t^2<br />

    The list goes on and on.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    Thanks for the reply Danny. Unfortunately, I was not given any BC's or IC's, I was just told to determine the number of solutions to the equation Ut = Uxx. As you stated above, I am guessing the answer would be infinite, but how do I know if it's infinitely many solutions or just a few. For example, how did you come up (or know) the list of solutions to the equation you listed in your reply above?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,367
    Thanks
    42
    Quote Originally Posted by spearfish View Post
    Thanks for the reply Danny. Unfortunately, I was not given any BC's or IC's, I was just told to determine the number of solutions to the equation Ut = Uxx. As you stated above, I am guessing the answer would be infinite, but how do I know if it's infinitely many solutions or just a few. For example, how did you come up (or know) the list of solutions to the equation you listed in your reply above?
    I've studied PDEs for more than twenty years and in that time I've read a lot and work alot with them. That's how I knew of these solutions. I also did some work about 10 years ago where we were able to show given one "seed" solution, we could generate a whole hierarchy of solutions. I can share that with you if you like.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    Wow, that's a lot of experience with PDE's!

    Thanks for the offer Danny. Although sharing this info with me would be nice, I am afraid I would be lost, since I am just beginning my journey to understanding these PDE's. What about solutions in the form of U(x,t) = exp^(ax + bt). How would I go about finding solutions of this form or tell how many solns of this form it has? Last question, I promise, lol.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,367
    Thanks
    42
    Quote Originally Posted by spearfish View Post
    Wow, that's a lot of experience with PDE's!

    Thanks for the offer Danny. Although sharing this info with me would be nice, I am afraid I would be lost, since I am just beginning my journey to understanding these PDE's. What about solutions in the form of U(x,t) = exp^(ax + bt). How would I go about finding solutions of this form or tell how many solns of this form it has? Last question, I promise, lol.
    Actually, just substitute this form into the PDE and require it be automatically satisfied. So here

    u_t = b e^{ax + bt},\;\;\;u_{xx} = a^2 e^{ax+bt}

    giving u_t = u_{xx}\;\;\; \Rightarrow\;\;\; b e^{ax + bt} = a^2 e^{ax+bt}

    So what's the condition on a and b such that this is satisfied?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    There were no conditions on a and b. The problem just said to find as many solutions for Ut = Uxx in the form of

    <br />
U_{x,t} = e^{ax + bt}<br />
.

    No other information was given.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,367
    Thanks
    42
    Quote Originally Posted by Danny View Post
    Actually, just substitute this form into the PDE and require it be automatically satisfied. So here

    u_t = b e^{ax + bt},\;\;\;u_{xx} = a^2 e^{ax+bt}

    giving u_t = u_{xx}\;\;\; \Rightarrow\;\;\; b e^{ax + bt} = a^2 e^{ax+bt}

    So what's the condition on a and b such that this is satisfied?
    Let me help a bit more. From

    b e^{ax + bt} = a^2 e^{ax+bt} we see that this is only satisfied (for nonzero a and b) if b = a^2. Thus, we have solution of the form

     <br />
u(x,y) = e^{a x+a^2 t}<br />
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    Danny,

    Once again, I can't thank you enough for taking the time to help me understand. I have a better understanding now. I am sure I ll have plenty more questions as my study of these PDE's progresses, but I ll jut take it one step at a time for now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. solving for x
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: November 7th 2009, 04:13 PM
  2. help solving for y
    Posted in the Algebra Forum
    Replies: 2
    Last Post: November 1st 2009, 10:25 AM
  3. help me in solving this
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 13th 2008, 01:51 PM
  4. Solving x'Px = v
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 11th 2008, 03:21 PM
  5. Replies: 3
    Last Post: October 11th 2006, 09:15 PM

Search Tags


/mathhelpforum @mathhelpforum