# Non-linear non-constant coefficient second order ODE

1. Nov 23, 2015

### Curious me

I would like to solve the steady-state one dimensional heat equation for a two piece material system. The thermal conductivity in each segment is a linear function of temperature, where $\kappa_1=a_1T+b_1$ for material 1 and $\kappa_2=a_2T+b_2$ for material 2. $a_1, a_2, b_1, and \;b_2$ are constants and T is temperature. Essentially $\kappa$ depends on both temperature and space since we have two materials.

I will explain my approach to solving this. I am hoping to see what you think about its correctness and if you identify it as an approach with a specific name that I am unaware of. Thank you. I should specify that using the described method I have obtained an analytical solution in Mathematica.

First, solve $\frac{\partial}{\partial x}(\kappa\frac{\partial T}{\partial x})=0$. This will give you a temperature profile as a function of x with two unknown coefficients that are to be determined by the boundary conditions.

Then substitute $\kappa$ with $\kappa_1\; and\; \kappa_2$ to obtain two separate temperature profiles for each section. Now you will have to determine four coefficients. To do this use the following boundary conditions:
$1)\; T_i=T_h\;$
$2)\; T_f=T_c$ where $T_h$ and $T_c$ are known.
$3)\;\kappa_1\frac{\partial T_1}{\partial x}=\kappa_2\frac{\partial T_2}{\partial x}$ (constant heat flux)
4) the boundary temperatures equate.

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Thank you

2. Nov 28, 2015