Nonclassical Symmetries for a Class of Reaction-Diffusion Equations: the Method of Heir-Equations

Abstract

The nonclassical symmetries method is applied to a class of reaction-diffusion equations with nonlinear source, i.e. u t = u xx + cu x + R (u, x). Several cases are obtained by using suitable solutions of the heir-equations as described in [M.C. Nucci, Nonclassical symmetries as special solutions of heir-equations,

Differential Equations Invariant Under Conditional Symmetries

“…There are several works devoted to using the non-classical method to construct solutions of PDEs that are different from the ones obtained by classical method using the Lie point symmetries. Among them, let us cite as an example, [2,3,10,16,20,21,24,33,36]. In the case of integrable equations let us mention the works of Sergyeyev [38,39] where he considered the classification of all (1 + 1)-dimensional evolution systems that admit a generalized (Lie-Bäcklund) vector field as a generalized conditional symmetry.…”

Lie group analysis for initial and boundary value problem of time-fractional nonlinear generalized KdV partial differential equation

“…Further, as our equation (1.1) does not involve the space-fractional derivative with respect to x , the conservation law for the x -component has the form [4] 6) where…”

A Note on the Equivalence of Methods to finding Nonclassical Determining Equations

“…In particular we address the problem posed by Hashemi and Nucci [4] who considered equations of the form (1.1): "We hope that an independent researcher will take up the task of comparing the two methods [of Bîlã and Niesen and Nucci] since we conjecture that Bîlã and Niesen's method, and its extension, as given in [2], are equivalent to Nucci's method. "…”