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

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

“…When 𝑓 (u), g(u), h(u), l(u), p(u), q(u), and k(u) are determined as some special functions, using the invariant subspace method, various exact solutions of Equation (1.1) were investigated by Prakash et al [35]. However, when…”

“…And this kind of nonlinear ordinary differential equation is just so suitable to use this integral bifurcation method to obtain its various exact solutions. To test the effectiveness of this improved method, we will investigate exact solutions and their dynamic properties of a class of time-fractional generalized thin-film equations [35] formed as follows:…”

“…And this kind of nonlinear ordinary differential equation is just so suitable to use this integral bifurcation method to obtain its various exact solutions. To test the effectiveness of this improved method, we will investigate exact solutions and their dynamic properties of a class of time‐fractional generalized thin‐film equations [35] formed as follows: $$$$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial^{\alpha }u}{\partial {t}^{\alpha }}=& -f(u){u}_{xx x x}-g(u){u}_x{u}_{xx x}-h(u){\left({u}_{xx}\right)}^2-l(u){\left({u}_x\right)}^2{u}_{xx}\hfill \\ {}\hfill & +p(u){u}_{xx}+q(u){\left({u}_x\right)}^2+k(u),\hfill \end{array}} $$$$ where $$$ \frac{\partial^{\alpha }}{\partial {t}^{\alpha }} $$$ can be Riemann–Liouville or Caputo fractional differential operator and $$$ u=u\left(x,t\right),x\in \mathrm{\mathbb{R}},t>0,0<\alpha \le 1 $$$. In particular, when …”

“…When $$$ f(u),g(u),h(u),l(u),p(u),q(u) $$$, and $$$ k(u) $$$ are determined as some special functions, using the invariant subspace method, various exact solutions of Equation () were investigated by Prakash et al [35]. However, when $$$ f(u)=\eta u,g(u)=3\eta, h(u)=2\eta, l(u)=0,p(u)=2\beta u,q(u)=2\beta, k(u)=\delta u $$$ or $$$ f(u)=\eta u,g(u)=4\eta, h(u)=3\eta, l(u)=0,p(u)=2\beta u+\delta, q(u)=2\beta, k(u)=0 $$$, exact solutions of Equation () are not investigated.…”

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.…”