Symmetries and solutions to the thin film equations
The author considers the determination of solutions by symmetry techniques of the nonlinear \(1 +1 \) fourth-order partial differential equation \[ u_t = -f (u)u_{xxxx} - g (u)u_xu_{xxx} - h (u)u_{xx} ^2 - d (u)u_x ^2u_{xx} + p (u)u_{xx} + q (u)u_x ^2 \tag{1} \] for certain smooth functions \(f \), \(g \), \(h \), \(d \), \(p \) and \(q \). Equation (1) for specific coefficient functions arises in the modelling of thin films and other fluid-mechanical applications. The author approaches the symmetry analysis of (1) with the assumption of specific classes of symmetry rather than the method of preliminary group classification which would seem to be a more natural approach since the assumption of a particular symmetry is more a matter of the taste of the assumer rather than the dictate of the nature of the equation. However, the author does make an interesting extension to the standard assumptions by the use of nonlinear extensions of the scaling group as introduced by Galaktionov. Consequently a variety of closed-formed solutions of (1) is obtained. Of the many cases considered some display divergence in finite time. It is probable that there are other classes of functions in (1) for which closed-formed solutions can be obtained by means of symmetry analysis. It would be interesting to see a singularity analysis would lead to different classes of solutions for this could give a pointer to further Ansätze for the proper choice of symmetries. The reader is advised that in the same context prime is used to denote differentiation with respect to different variables.
scientific article; zbMATH DE number 5017388
scientific article