Invariant subspaces and exact solutions: (1+1) and (2+1)-dimensional generalized time-fractional thin-film equations

We investigate the applicability and efficiency of the invariant subspace method to (2 + 1)-dimensional time-fractional nonlinear PDEs. We show how to find various types of invariant subspaces and reductions for the (1 + 1) and (2 + 1)-dimensional generalized nonlinear time-fractional thin-film equations which arise from the motion of liquid film on a solid surface under the influence of surface tension. We construct several kinds of exact solutions for the above-mentioned equations depending on arbitrary functions as either a combination of trigonometric, polynomial, Mittag-Leffler, and exponential type functions or any of these forms. Also, we demonstrate the applicability of the invariant subspace method to solve the initial and boundary value problem of nonlinear time-fractional PDEs for the first time and illustrate it through the physically important generalized time-fractional thin-film and linear time-fractional heat equations. Finally, we present some of the obtained exact solutions graphically.