Group Transformations and the Nonlinear Heat Diffusion Equation

A quantum particle in a box with moving walls

“…In particular, the solvability of the model was investigated in a series of papers by Dembiński, Makowski and others [5][6][7][8] who exhibited several exact and perturbative solutions. Moreover, Dodonov et al [9] analyzed uniformly moving walls, including the cases of adiabatic motion and almost sudden changes of the wall position, while Munier et al [10] and Konishi and Paffuti [11] investigated the behavior of the average energy. The problem in higher dimensions was considered by Lenz et al [12] and by Del Campo and Boshier [13], who focused on BoseEinstein condensates and experimental realizations.…”

Similarity solutions of reaction–diffusion equation with space- and time-dependent diffusion and reaction terms

“…From W (x, t) = t µ y(z) and µ = −α = −1/(n + 2), we arrive easily at the solution given in [16] W (x, t) = Ct…”

Exact solutions of diffusion-convection equations

“…Potential equivalence transformations were used to obtain solutions of some boundary-value problems adduced in [10,41,63,64] 4 Nonlinear diffusion equations. General case Consider now the class of nonlinear diffusion equations…”