A new regularisation for time-fractional backward heat conduction problem

It is well-known that the backward heat conduction problem of recovering the temperature $u(\cdot ,t)$ at a time $t\ge 0$ from the knowledge of the temperature at a later time, namely $g:=u(\cdot ,\tau )$ for $\tau >t$, is ill-posed, in the sense that small error in $g$ can lead to large deviation in $u(\cdot ,t)$. However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for $t>0$ and ill-posed for $t=0$. We use this observation to obtain stable approximate solutions for the TFBHCP for $t=0$, and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.