A group analysis approach for a nonlinear differential system arising in diffusion phenomena

Abstract

We consider a class of second-order partial differential equations which arises in diffusion phenomena and, following a new approach, we look for a Lie invariance classification via equivalence transformations. A class of exact invariant solutions containing an arbitrary function is obtained.

Exact solutions of a reaction–diffusion system for Proteus mirabilis bacterial colonies

“…We apply, once having obtained the continuous equivalence infinitesimal generator Y of the class (1), the following projection theorem that allows us to get the symmetry X by projecting Y in the space {t, x, u, v}. This theorem has been introduced in [14] and eventually reconsidered in [7,12,13,8].…”

“…At this step, it is useful in view of further applications, to not require the invariance of the auxiliary conditions [12,13,8]…”

Investigation of Ultrasonic Motors Using Thin Piezo-Ceramics and a Metal Composite-Plate for Watches

“…The same strategy is followed when WETs are employed [9][10][11][12] but this latter procedure, as shown in [1,[9][10][11][12], usually gives a wider set of symmetries. A procedure based on equivalence or WETs does not, in general, ensure determination of the complete symmetry classification, but in applications it reveals a successful and computationally appealing way to obtain symmetries.…”

“…However, since in (1), (2), the functions, not a priori assigned, p 1 , p 2 and p 3 appear, the computational difficulties considerably increase and it is too involved to get the complete symmetry classification by the Lie direct method. Therefore, it is convenient to proceed to look for equivalence transformations or weak equivalence transformations as in [7][8][9][10][11][12] where the problem of symmetry classification in the presence of arbitrary functions has been considered for different physical models.…”

Lie group classification of the N-th-order nonlinear evolution equations

“…Note that equivalence group of Equation (1) is the Lie transformation group which preserves the class of NLEEs (1). In the second step, based on the explicit forms of commutation relations for low dimensional abstract Lie algebras [20][21][22][23][24], we provide all inequivalent realizations of symmetry algebras by basic operators admitted by Equation (1). Finally in the last step, inserting the canonical forms of symmetry generators into the classifying equations and solving them, we derive the explicit forms of invariant equations.…”