Invariant analysis, exact solutions, and conservation laws of time fractional thin liquid film equations

Abstract

This present paper investigates Lie symmetry analysis, one-dimensional optimal system, exact solutions and conservation laws of the (2 + 1)-dimensional time fractional thin liquid film equations (TFTLFE) with Riemann–Liouville fractional derivative. Explicitly, we obtain six vector fields and the one-dimensional optimal system admitted by TFTLFE. Then, we perform the symmetry reductions with the help of Erdélyi–Kober fractional differential operator and (2 + 1)-dimensional TFTLFE is reduced into (1 + 1)-dimensional fractional partial differential equations (FPDEs). Additionally, by means of compound variable transformation and the power series expansion method, the solution of reduced FPDEs is obtained and its convergence is verified. Moreover, we derive other solutions for the reduced equations taking advantage of the invariant subspace method. Furthermore, the conservation laws are also established utilizing generalized Noether's theorem. Finally, we construct the exact solution using the method of conservation laws.

Invariant Subspaces of Short Pulse-Type Equations and Reductions

“…The invariant subspace method is powerful for studying nonlinear partial differential equations (PDEs). Various invariant subspaces to a number of nonlinear PDEs have been obtained (see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], as well as the references therein). Accordingly, exact solutions stemming from this method play important roles in the study of their asymptotical behavior, blow up and geometric properties, etc.…”

“…Let us introduce the invariant subspace method briefly [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Consider the following nonlinear PDEs…”