  In this paper, we obtain a Carleman estimate for the higher order partial
differential operator. In the process of establishing this estimate, we
developed a new method, which is called the back-propagation method (the BPM,
for short). This method can also be used to build up Carleman estimates for
some other partial differential operators, and might provide assistance with
corresponding numerical analyses. As an application of the above-mentioned
Carleman estimate, we proved the conditional stability of a Cauchy problem for
a time fractional diffusion equation.
