Solving a 1D heat transfer ODE BVP problem $\frac{d}{dx}(k\frac{du}{dx})$ over three homogeneous layers.

Question

Question

I need help on part a of this question

Context

• This is from a book on the finite element method. Finite elements a gentle introduction Chapter 4 question 5
• From the answer at the back of book the true solution satisfies the following coordinate pairs $(p_1,70), (p_2,50),(p_3,46)$. (Note the true solution is not given, just the values of $u$ at each boundary between the layers).
• Below I have given my solution for part a, I need feedback to understand where I went wrong.
• If you need any clarification feel free to ask.

Work thus Far

My approach is to solve a BVP, treating each layer as separate boundary value problem, so that $k$ is treated as a constant and not a piecewise function. So \begin{align} \frac{d}{dx}(k\frac{du}{dx})&=\frac{d^2 u}{dx^2}=0. \end{align} and has the solution $u(x)=c_1 x+c_2$.

In order to satisfy the left most boundary condition $u(0)=c_2=50$. To determine $c_1$ we need to look at the left boundary of layer $w_1$. Note that as the next layers do not impose conditions on $c_1$, $c_2$ must change. In order to avoid confusion I relabel the equation to $$u(x)=\bar{c}_i x+50$$. In order for the boundary conditions to be satisfied we need the following to hold \begin{align} k_1\frac{d u}{dx}&=k_2 \frac{du}{dx}\\ k_2\frac{d u}{dx}&=k_3 \frac{du}{dx}\\ k_3\frac{d u}{dx}&=-h(u(b)-u_{\infty}) \end{align} or \begin{align} k_1 \bar{c}_1&=k_2 \bar{c}_2\\ k_2 \bar{c}_2 &=k_3 \bar{c}_3\\ k_3 \bar{c}_3&=-h(u(b)-26) \end{align}.

Assuming $u(b)=\bar{c}_3 x +50$ then using the last equation, $c_3=-800/9$. This must be incorrect as that means $u(b)=12.4$ and not $u(b)=46$ (which is what the answer says). It would also mean all $\bar{c}_i$ are negative which does not make sense given that $u(0)=50$ (which is given) and $w(p_1)>50$ (which is implied by the answer).

What am I doing wrong?

Answer 1

Since $k$ is piecewise constant, the solution is piecewise linear. So you have $$ u(x) = \begin{cases} c_1x + d_1, & 0 < x < w_1 \\ c_2x + d_2, & w_1 < x < w_2 \\ c_3x + d_3, & w_2 < x < w_3 \end{cases} $$

There are 6 unknown constants here, so 6 conditions are needed to determine them. Two of them are conditions on the endpoints, two of them are continuity on $u(x)$, and two of them are continuity on $ku'(x)$. In total:

\begin{align} d_1 &= u^* \\ c_1w_1 + d_1 &= c_2w_1 + d_2 \\ c_2w_2 + d_2 &= c_3w_2 + d_3 \\ k_1c_1 &= k_2c_2 \\ k_2c_2 &= k_3c_3 \\ -k_3c_3 &= h(c_3w_3 + d_3 - u_{\infty}) \end{align}

where $k_i$, $w_i$, $u^*$, $u_\infty$ and $h$ are known constants.