Nonclassical Symmetry Analysis Research Articles

Non-Classical Symmetry Analysis of a Class of Nonlinear Lattice Equations

In this paper, a non-classical symmetry method for obtaining the symmetries of differential–difference equations is proposed. The non-classical symmetry method introduces an additional constraint known as the invariant surface condition, which is applied after the infinitesimal transformation. By solving the governing equations that satisfy this condition, we can obtain the corresponding reduced equation. This allows us to determine the non-classical symmetry of the differential–difference equation. This method avoids the complicated calculation involved in extending the infinitesimal generator and allows for a wider range of symmetry forms. As a result, it enables the derivation of a greater number of differential–difference equations. In this paper, two kinds of (2+1)-dimensional Toda-like lattice equations are taken as examples, and their corresponding symmetric and reduced equations are obtained using the non-classical symmetry method.

Conditionally integrable PDEs, non-classical symmetries and applications

Some multi-dimensional nonlinear partial differential equations reduce to a linear equation in fewer dimensions after imposing one additional constraint. A large class of useful conditionally integrable reaction–diffusion equations follows from a single non-classical symmetry reduction, yielding an infinite-dimensional but incomplete solution space. This solution device extends further to other nonlinear equations such as higher-order reaction–diffusion, fractional-order diffusion and diffusion–convection. Applications are shown for population dynamics with or without weak Allee effects, speed-limited hyperbolic diffusion, material phase field dynamics, soil–water–plant dynamics and calcium transport on the curved surface of an oocyte. Beyond non-classical symmetry analysis, examples of other conditionally integrable equations are given; nonlinear diffusion in 1 + 2 dimensions and the Madelung–Burgers quantum fluid in 1 + 3 dimensions.

Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

Exact time-dependent solutions of a Fisher–KPP-like equation obtained with nonclassical symmetry analysis

We consider a family of exact solutions to a nonlinear reaction–diffusion model, constructed using nonclassical symmetry analysis. In a particular limit, the mathematical model approaches the well-known Fisher–KPP model, which means that it is related to various applications including cancer progression, wound healing and ecological invasion. The exact solution is mathematically interesting since exact solutions of the Fisher–KPP model are rare, and often restricted to long-time travelling wave solutions for special values of the travelling wave speed.

Nonclassical symmetry analysis, conservation laws of one-dimensional macroscopic production model and evolution of nonlinear waves

In this article we consider a system of first order quasilinear hyperbolic partial differential equations (PDEs) governing the one-dimensional macroscopic production model which describes high-volume product flows. We first show that the governing system admits nonlinear self-adjointness property. Further, we construct various conservation laws of the given system by using the direct multipliers method and also each of its classical symmetries yields a conservation law of the original system due to Ibragimov's formulation. Then we perform the nonclassical symmetry analysis and compute several hidden symmetries. Using these nonclassical symmetries, we obtain some new exact solutions and discuss their physical behavior graphically. Finally, the evolution of characteristic shock and weak discontinuity are studied by using one of the obtained solutions.

Nonclassical Symmetry Analysis of Heated Two-Dimensional Flow Problems

Abstract This article analyses the nonclassical symmetries and group invariant solution of boundary layer equations for two-dimensional heated flows. First, we derive the nonclassical symmetry determining equations with the aid of the computer package SADE. We solve these equations directly to obtain nonclassical symmetries. We follow standard procedure of computing nonclassical symmetries and consider two different scenarios, ξ 1≠0 and ξ 1=0, ξ 2≠0. Several nonclassical symmetries are reported for both scenarios. Furthermore, numerous group invariant solutions for nonclassical symmetries are derived. The similarity variables associated with each nonclassical symmetry are computed. The similarity variables reduce the system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) in terms of similarity variables. The reduced system of ODEs are solved to obtain group invariant solution for governing boundary layer equations for two-dimensional heated flow problems. We successfully formulate a physical problem of heat transfer analysis for fluid flow over a linearly stretching porous plat and, with suitable boundary conditions, we solve this problem.

Symmetry analysis of a heat conduction model for heat transfer in a longitudinal rectangular fin of a heterogeneous material

We consider a heat conduction model arising in transient heat transfer through longitudinal fins of a heterogeneous (functionally graded) material. In this case, the thermal conductivity depends on the spatial variable. The heat transfer coefficient depends on the temperature and is given by the power law function. The resulting nonlinear partial differential equation is analyzed using both classical and nonclassical symmetry techniques. Both the transient state and the steady state result in a number of exotic symmetries being admitted by the governing equation. Furthermore, nonclassical symmetries are also admitted. Both classical and nonclassical symmetry analysis results in some useful reductions and some remarkable exact solutions are constructed.

Nonclassical Symmetry Analysis of Boundary Layer Equations

The nonclassical symmetries of boundary layer equations for two‐dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

Nonclassical symmetry of nonlinear diffusion equation and factorization method

We present the nonclassical symmetry of a nonlinear diffusion equation whose a nonlinear term is an arbitrary function. Generally, there is no guarantee that we can always determine the nonclassical symmetries admitted by the given equation because of the nonlinearity included in the determining equations. Accordingly, constructing invariant solutions is also generally difficult. In this paper, we apply the factorization method to nonclassical symmetry analysis for the nonlinear diffusion equation. Applying this method simplifies the determining equations and leads to their invariant solution automatically.

Nonclassical symmetry analysis for hyperbolic partial differential equation

We consider the nonclassical symmetry of one-dimensional hyperbolic differential equations of the form u t + M( u) u x = 0. For the infinitesimal generator V = τ ∂ t + ξ ∂ x + ∑ i = 1 n ϕ i ∂ u i , it is shown that ξ is an eigenvalue of the matrix M when ϕ i = 0 [Souichi M. Nonclassical symmetry and Riemann invariants. Int J Nonlinear Mech, [in press]]. In this paper, we prove a sufficient condition of a lemma.

Similarity solutions of an equation describing ice sheet dynamics

This paper focuses upon the derivation of the similarity solutions of a nonlinear equation associated with a free boundary problem arising in glaciology. We present a potential symmetry analysis of this second order nonlinear degenerate parabolic equation related to non-Newtonian ice sheet dynamics in the isothermal case. A full classical and also a non-classical symmetry analysis are presented. After obtaining a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation (without any unilateral formulation), a particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This allows us to obtain several qualitative properties of the free moving boundary in the presence of an accumulation–ablation function with realistic physical properties.

The coupled modified Korteweg–de Vries equations: Similarity reduction, Lie–Bäcklund symmetries and integrability

The Lie point symmetries of N-coupled modified Korteweg–de Vries equations (N-cmKdV) is derived and shown that the similarity reduction associated with symmetries passes the Painlevé property indicating its integrability. The nonclassical symmetry analysis of Bluman and Cole and the direct method of Clarkson and Kruskal to N-cmKdV equations show that there exists no new similarity reductions. A sequence of Lie–Bäcklund symmetries for N-cmKdV equations is derived explicitly, establishing its complete integrability. The question of constructing the recursion operator of N-cmKdV, a characteristic of integrable systems, is also discussed.