Symmetries of the heat equation on the lattice (Q1912758)

The symmetry operators of the discrete version (1) \([\Delta_t- \Delta^2_x ]\varphi (t, x)= 0\), \((t, x)\in \mathbb{R}^2\) of the heat equation (HE) in one space dimension are presented. Here \(\Delta_a= {1\over C_a} [T_a- 1]\), \(T_a\) are the shift operators and \(C_a\) are the lattice constants. It turns out that this partial difference equation has the same symmetry algebra as its continuous limit -- the usual HE. Finally, this symmetry algebra is exploited to find some solutions of (1) and, next, to study their properties. Note also a similar study of the \(q\)-difference analog of the HE carried out by \textit{R. Floreanini} and \textit{L. Vinet} [Lett. Math. Phys. 32, No. 1, 37-44 (1994; Zbl 0805.39008)].