Application of Lie point symmetries to the resolution of an interface problem in a generalized Fisher equation

Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

“…This NLODE (8) was integrated directly and its solution was obtained in the form of an incomplete elliptic integral. Moreover, Kudrayshov's method was employed to obtain the solution of the NLODE (8). These solutions were presented graphically.…”

“…The first step is to reduce the NPDE (1) to a nonlinear ODE, which we already performed using the Lie symmetries in the previous section. Thus, we work with the ODE (8). We suppose that a solution of (8) can be expressed as…”

“…In this subsection, we derive a solution of the two-dimensional generalized shallow water Equation ( 1) by direct integration of the ODE (8). Taking U (q) = V (q), Equation ( 8) becomes…”

“…In addition, the Boussinesq-Burgers-type system of equations, which delineates shallow water waves appearing close to lakes or ocean beaches, was studied in [4]. The list continues; see also [5][6][7][8][9][10][11][12][13][14][15].…”

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

“…Furthermore, Lie's classical theory represented a source for various generalizations. Among these generalizations there are the nonclassical symmetries first proposed by Bluman and Cole [43], and now part of the more general method of differential constraints [44,45], the potential symmetries [46], the nonlocal symmetries [47][48][49], the gen-eralized symmetries [5], which in turn generalize contact symmetries introduced by Lie himself, the equivalence transformations [3,[50][51][52][53][54][55][56], to quote a few. A further extension is represented by approximate symmetries [57][58][59][60][61] for differential equations containing small terms, often arising in concrete applied problems (see, for instance, [62] for an application of approximate Lie symmetries to Navier-Stokes equations).…”