Exact solutions of reaction-diffusion-convection equations and their applications

Plane deformations of the Varga material

“…where X , Y , U , V , R , and S , the components of the infinitesimal transformations which are functions of x , y , u , v , r , and s . The procedure is very well-documented, and we refer the reader to several books on the subject matter (see, for example, [10–14]). Performing this analysis leads to a total of 50 determining equations several of which are duplications or linear combinations of others.…”

The discretized Boussinesq equation and its conditional symmetry reduction

“…where the expression C = 0 represents the condition and all of its differential consequences up to the order of the equation, see [12,18] and the references therein for more results.…”

“…They do not form a group but they can be used to obtain additional reductions leading to explicit solutions. Conditional symmetries and the reductions provided by them have been applied to obtain explicit (new, physically relevant) solutions for problems arising in non-newtonian fluid flow models [1], in magnetohydro dynamics [20], in reaction-diffusion-convection equations [12], in equations of heat and acoustics [18], etc.…”

Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems

“…The notions of conditional and nonclassical symmetries, originating most prominently in the works of Bluman and Cole [2], Fushchych [20,21], Olver [33], Winternitz [28] and their co-authors, eventually opened up the possibility of new reductions and solutions to practical partial differential equations that could not be obtained by Lie's classical algorithm. Notably, techniques like the method of differential constraints of Yanenko [38], the direct reduction method of Clarkson and Kruskal [18] and some others (see for details Chapter 5 of book [17]), which are not based on symmetries, also can help in constructing new non-Lie solutions for nonlinear PDEs arising in real-world applications.…”