Lie symmetries and exact solutions for a fourth‐order nonlinear diffusion equation

Abstract

In this paper, we consider a fourth-order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one-dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one-dimensional subalgebras.Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

“…This section introduces conservation laws for KdV-mKdV equation associated with Lie symmetry analysis [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. The adjoint equation and Lagrangian operator are determined using Noether's theorem.…”

“…Therefore, nonlinear partial differential equations (NPDEs) play a major role in modeling of these natural waves and several wave phenomena as well. The massive applications of NPDEs can be seen in various fields of mathematical sciences, biological sciences, nonlinear optics, electromagnetic theory, quantum theory, optical fiber, plasma physics, heat transfer, fluid dynamics, and so forth [1‐32,33,34]. The NPDEs are difficult to handle with traditional methods, so a large number of methods such as Darboux transformations [1], Fan's subequation method [3], extended mapping method [4], sub‐ODE method [5], extended tanh method [6], amplitude ansatz method [7], WTC truncation method [8], nonlocal symmetry method [9], and similarity transformation method [10–34] are evolved to derive exact solutions.…”

“…KdV‐mKdV equation is widely applied in several contexts as density stratified ocean, impact of external forces, internal gravity waves, dust‐ion acoustic waves, traffic flow problems, and so forth [1–10]. To accomplish the exact solution of governing equation, authors are motivated from previous researches to employ similarity transformation method via Lie group theory and derive conserved quantities [1‐32,33,34]. The method depends on invariance property under one parameter transformation.…”

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

“…The Lie group approach provides a powerful framework for analyzing partial differential equations, as it enables their invariance by employing geometric infinitesimal transformations on an underlying manifold. Due to this fact, this method has been extensively used to construct some invariant solutions and conservation laws for some equations [12][13][14]. Moreover, some extensions of the Lie transformation groups have been proposed, such as building nonclassical symmetries [15] and nonlocal residual symmetries [16] for partial differential equations and systems.…”

Lie Symmetry Analysis and Conservation Laws of the Axially Loaded Euler Beam

“…Ref. [26] also constructed the exact solutions of fourth-order diffusion equations by using the Lie symmetry method. Although it did not refer to the conservation laws, it was of significant reference for the symmetry reduction of the fourth-order Euler beam.…”