I was reading a paper about one-dimensional heat conduction problem and I get stuck in one expression that I couldn't understand how to calculate.
First, they define the heat conduction problem as follows:
$\frac{\partial u}{\partial t} = \frac{\partial}{\partial t}[q(x)\frac{\partial u}{\partial x}]+f(x,t),\quad (x,t) \in (0,L) \times (0,T]$
with initial condition $u(x,0) = u_{0}(x)$, $0\leq x\leq L$
and Dirichlet boundary conditions
$u(0,t) = g_{1}(t), 0\leq t\leq T$,
$u(1,t) = g_{2}(t), 0\leq t\leq T$,
where $f(x,t), u_{0}(x), g_{1}(t), g_{2}(t)$, and $q(x)$ are known continuous functions. This is considered as a direct problem. Also the diffusion coefficient is known and is given by $q(x)= p_{1}+p_{2}x+...+p_{m+1}x^{m} $, where $p_{1}..., p_{m+1}$ are constants.
For the inverse problem, the $q(x)$ is unknown. In order to estimate it they said that additional information on the boundary $x=x_{0}, 0<x_{0}<L,$ is required and they denote $u(x_{0},t) = g(t) , 0\leq t\leq T$. Until now everything is clear.
They said that to find $q(x)$ one has to determine its coefficients $p_{1},p_{2}..., p_{m+1}$ by minimizing the error estimate (using least-squares method) :
$F(p_{1},...p_{m+1}) = \sum_{i=1}^{n} [u(x_{0},t_{i},p_{1},p_{2}...,p_{m+1})-g(t_{i})]^{2}$.
What i didn't understand is how can i define the expression of $u(x_{0},t_{i},p_{1},p_{2}...,p_{m+1})$ ? They mentioned that these are the calculated results and are determined from the solution of the direct problem which is given previously by using $q(x)$, but i really don't know how can i derive this expression. Note that in the forward problem one can calculate the solution $u(x,t)$ as well as $q(x)$.
I would be very thankful for your explanations !
You can find the paper here https://www.hindawi.com/journals/mpe/2014/626037/
If you fix a value for $p_1,\ldots,p_{m+1}$, for example $p_1 = p_2=\ldots=p_{m+1} = 1$, then you can solve the direct problem explicitly. So $u(t,x,1,\ldots,1) = u(t,x)$, where $u(t,x)$ is the solution of the direct problem with $q(x) = 1 + x + \ldots + x^{m}$.
To put it differently: in the simplified case $m=0$ we have $q(x) = p_1$. Then to see what $u(t,x,p_1)$ is, you solve the equation $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}[p_1\frac{\partial u}{\partial x}] + f(x,t)$ (with the previous boundary/initial data). You will then get an explicit form of $u$ which somehow depends on $p_1$. Then you do the least-square optimization to see which $p_1$ works best.