A new regularization for time-fractional backward heat conduction problem

Abstract

It is well known that the backward heat conduction problem of recovering the temperature u ⁢ ( ⋅ , t ) at a time t ≥ 0 from the knowledge of the temperature at a later time, namely g := u ⁢ ( ⋅ , τ ) for τ > t , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ ( ⋅ , 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 as solutions for t ∈ ( 0 , τ ] with t as regularization parameter for approximating the solution at 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.