Fitting an experimental data to the finite difference approximated solution

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Hi, I know the finite difference approximation is well established. However, I have one question. I am working on a simple system where I measured temperature at the boundary of that particular system. Say, left boundary is maintained at a constant temperature and right one has no flux condition. I have experimental values of temperature over the time period at right boundary. I am following FTCS method and I am trying to get the profile at the boundary by changing thermal conductivity value. I am just wondering if anyone can help me out to get approximated thermal conductivity using experimental data.

Thanks