A hybrid regularization method for identifying the source term and the initial value simultaneously for fractional pseudo-parabolic equation with involution

The article explores the inverse problem of simultaneously identifying the source term and initial value for a fractional pseudo-parabolic equation with involution. It introduces a hybrid regularization method that combines modified quasi-inverse and quasi-initial value approaches to address the ill-posed nature of the problem. The method is validated through theoretical analysis, including conditional stability results and convergent error estimations under both a priori and a posteriori regularization parameter selection rules. Numerical examples demonstrate the effectiveness and accuracy of the proposed method, highlighting its potential for practical applications in fields such as population dynamics, fluid dynamics, and game theory. The article concludes by suggesting future research directions, including the application of other regularization methods for comparative analysis.