Results 1 to 4 of 4

Math Help - Formulating the BV problem

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Formulating the BV problem

    In a rod length L with insulated lateral surface and thermal constants c, \ \rho, \ \kappa, \ K heat is generated at a rate uniformly proportional to the temperature, that is, at a rate ru(x,t) per unit volume per unit time, where r is a constant and u(x,t) is the temperature function of the rod. The ends of the rod are maintained at temperature 0 and initially the rod has uniform temperature 1.

    Formulate an initial-BV problem for determining u(x,t)

    \text{D.E.}: \ u_t=ru_{xx}
    \displaystyle\text{B.C.}=\begin{cases}u(0,t)=0\\u(  L,t)=0\end{cases}
    \text{I.C.}: \ u(x,0)=1

    I am not sure what to do with the r so I just put it in front of u_{xx}. Is that correct? Also, I need a K, c, and rho somewhere but not sure where and why.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    r does belong there. It is known as Thermal Diffusivity.

    It relates thermal conductivity and volumetric heat capacity together. It specifically shows how conductivity changes in comparison to the thermal bulk. As to why it belongs there, I don't know. Maybe seeing the equation will help you figure it out.

    r = \frac{K}{\rho c_p}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by Aryth View Post
    r does belong there. It is known as Thermal Diffusivity.

    It relates thermal conductivity and volumetric heat capacity together. It specifically shows how conductivity changes in comparison to the thermal bulk. As to why it belongs there, I don't know. Maybe seeing the equation will help you figure it out.

    r = \frac{K}{\rho c_p}
    But by making that substitution, I will lose r. In the final solution, there is an r, rho, K, and c.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    The DE is

    \displaystyle\text{D.E.}: \ u_t=ku_{xx}+\frac{r}{c\rho}u
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Formulating an expression
    Posted in the Algebra Forum
    Replies: 2
    Last Post: April 12th 2011, 09:21 AM
  2. Formulating a NLP
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 9th 2010, 06:47 PM
  3. Formulating a NLP
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 28th 2010, 03:24 AM
  4. formulating this problem..
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: October 17th 2009, 02:44 AM
  5. Formulating an Integer Programming problem
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: April 25th 2009, 01:11 AM

/mathhelpforum @mathhelpforum