This research work investigates the existence, uniqueness, and stability of solution for a time-fractional fourth-order partial differential equation, subject to two initial conditions and four nonlocal integral boundary conditions. The equation incorporates several key components: the Caputo fractional derivative operator, the Laplace operator, the biharmonic operator, as well as terms representing frictional and viscoelastic damping. The presence of these elements, particularly the nonlocal boundary constraints, introduces new mathematical challenges that require the development of advanced analytical methods. To address these challenges, we construct a functional analytic framework based on Sobolev spaces and employ energy estimates to rigorously prove the well-posedness of the problem.
Alrajhi et al. (Thu,) studied this question.