The modified fractional-order quasi-reversibility method for a class of direct and inverse problems governed by time-fractional heat equations with involution perturbation

This study aims to explore two classes of ill-posed problems governed by a nonclassical fractional heat equation with an involution perturbation. To achieve a stable solution, we introduce a modified pseudo-parabolic regularization method that involves a correction term through a mixed fractional derivation operator, resulting in a sequence of well-posed problems that depend on a regularization parameter epsilon$$ \varepsilon $$. We rigorously demonstrate that the resulting approximate problems are well-posed and present convergence results for this regularization approach.