A Homotopy Analysis Method for a Coupled Two-Dimensional Inverse Time-Fractional Navier–Stokes Problem with a Moving Boundary

Abstract

This study develops a semi-analytical framework for a coupled two-dimensional inverse time-fractional incompressible Navier-Stokes moving-boundary problem. The governing system consists of two coupled momentum equations involving a Caputo time-fractional derivative, nonlinear convective terms, a pressure field, viscous diffusion, and a time-dependent unknown coefficient multiplying prescribed spatial-temporal forcing functions. The problem is posed on a moving physical domain whose upper boundary is represented by an unknown interface. In addition to the initial and boundary conditions, an overdetermined condition is imposed to recover the unknown coefficient. The Homotopy Analysis Method (HAM) is used to construct approximate series solutions for the velocity components  and , while the pressure is recovered from the momentum equations, and the moving boundary is obtained through the vertical velocity condition on the interface. Exact solutions are used to validate the proposed method and to provide explicit benchmark expressions for the velocity field, pressure, moving boundary, source terms, and unknown coefficient. Numerical comparisons show that the second-order HAM approximation provides stable and accurate recovery of the horizontal velocity, pressure field, moving boundary, and unknown coefficient. Although the vertical velocity component exhibits larger relative errors due to the small magnitude of the exact solution, the approximation remains bounded and stable over the time interval. The results demonstrate that the proposed HAM-based approach provides an effective semi-analytical tool for solving coupled inverse fractional fluid-flow problems on moving domains.