Hint: We know that the backward heat conduction problem (shortly BHCP) is an extremely ill-posed problem in the sense of Hadamard. In fact, any small change in the terminal data can cause a dramatically large error in the solution. From the standpoint of the theoretical analysis, we can simply demonstrate that the BHCP is unstable. For verifying this fact form computational aspect, How can we implement it in the MATLAB or MATHEMATICA?
Dear Milad, interesting topic. See the attached paper, Chapter 3. Gianluca
I think that you can consider the classical numerical stability analysis methods. Therefore, a matrix analysis using Gershgorin locus or a Fourier analysis like the von Neumann. Have a look here http://mathcenter.hust.edu.cn/Uploads/file/20160128/56a98740d59f8.pdf
The key is that the problem is physically stiff as you want to describe a phenomenon where gradients are increased in time, this way small errors at high wavenumembers are amplified rapidily.
Hello Milad
For the BHCP the term 'unstable' is not necessarily appropriate. I'd rather say that it is an 'inverse problem' or it is 'ill-posed'.
If we solve the HCP forwards in time by a numerical method then the 'diffusion' that is inherent 'helps'. Ie any numerical error is diffused away. However, if we apply the same method backwards then the opposite happens; numerical error is magnified and such a method could be deemed 'unstable'. Hence lots of methods have arisen to develop a stable solution of solving the BHCP. I wrote a paper on one such method in 2002, but there are countless others.
Regards,
Stephen
Dear Milad
I think if you put some noise error (say Gaussian) in the input data you will obtain an unstable solution for ill-posed problem (in your case the BHCP). However, if not the problem will produce stable solution (in this case the problem is well-posed).
To overcome the first part the regularization needed in order to obtain stable solution. Of course with appropriate choice of regularization parameter(s).
Regards
Mohammed.
- Badges
- Science topic
- Similar topics
- Social Psychology
- Attitude