A novel regularization scheme for a backward problem in a sub-diffusion model

Abstract In this paper, an ill-posed problem of multi-term time-fraction diffusion equation with unknown initial values is considered. This is usually done by g := u ⁢ ( ⋅ , T ) {g:=u(\,\cdot\,,T)} for some T > 0 {T>0} to find u ⁢ ( x , t ) {u(x,t)} for 0 ≤ t < T {0\leq t which is so-called backward problem. For convenience, we will label this problem as ( P t ) {(P_{t})} . Through analysis, we know that ( P t ) {(P_{t})} is well posed for 0 < t < T {0 , but ill-posed for t = 0 {t=0} that is different from classical backward heat conduction problem. Therefore, a novel identification scheme of stable approximate solutions of ill-posed inverse problem ( P 0 ) {(P_{0})} is obtained by regularization family { R t : 0 < t < T } {\{R_{t}:0 , and the error estim