Find the steady-state temperature of the rod.

[eunice064]

A rod occupying the interval 0 ≤ x ≤ l is subject to the heat source

f(x) =0, for 0 < x < L/2
f(x) =H , for L/2 <x <L ,H>0

(1)The rod satisfies the heat equation ut = uxx + f(x) and its ends are kept at zero temperature. Find the steady-state temperature of the rod.
(2)Which pointis the hottest, and what is the temperature there?

From this point i don't know what to do

thanks

 

[LCKurtz]

A rod occupying the interval 0 ≤ x ≤ l is subject to the heat source

f(x) =0, for 0 < x < L/2
f(x) =H , for L/2 <x <L ,H>0

(1)The rod satisfies the heat equation ut = uxx + f(x) and its ends are kept at zero temperature. Find the steady-state temperature of the rod.
(2)Which pointis the hottest, and what is the temperature there?

From this point i don't know what to do

thanks

Are you sure you have stated the problem correctly? Isn't the 1-dimensional heat equation $u_t = u_{xx}$? And is that $f(x)$ you have given the initial condition $u(x,0)$? Why do you have $f(x)$ in the heat equation? Once you get that straight my hint would be to remember that steady state heat flow means there is no variation in temperature over time. So $u_t=0$.

 

[eunice064]

Are you sure you have stated the problem correctly? Isn't the 1-dimensional heat equation $u_t = u_{xx}$. And is that $f(x)$ you have given the initial condition $u(x,0)$? Why do you have $f(x)$ in the heat equation? Once you get that straight my hint would be to remember that steady state heat flow means there is no variation in temperature over time. So $u_t=0$.

thank you

so ut=0 ,i get uxx=f(x) with the boundary conditions is u(0)=0 & u(L)=0

How do I solve u?

 

[LCKurtz]

You might first address the issues I raised about the form of the heat equation and about f(x).

 

[LawrenceC]

You can break the domain into two parts, a left half and a right half. The left half is LaPlace equation in one dimension. The right half is the Poisson equation in one dimension. Solve them separately. The constant temperature boundary condition will make one constant of integration known for each domain. That leaves two unknown constants of integration, one for each half. They are determined by forcing continuity of temperature and heat flux at the midpoint which provides you with two equations so you can then determine the remaining two constants.