Numerical Simulation and Symmetry Reduction of a Two-Component Reaction-Diffusion System

In this paper, the symmetry classi ﬁ cation and symmetry reduction of a two-component reaction-di ﬀ usion system are investigated, the reaction-di ﬀ usion system can be reduced to system of ordinary di ﬀ erential equations, and the solutions and numerical simulation will be showed by examples.


Symmetry Reduction
In this section, we will illustrate the main feature of the reduction procedure. The systems (1) and (2) admit the conditional Lie-Bäcklund symmetry (CLBS) We mainly consider the following two cases.
Case 1. When b 1 = 0, then system can be derived to the following form and admits the CLBS: The systems (7) and (8) are a system of ordinary differential equations (ODEs) with respect to variable x, so the following forms are the corresponding solutions: In the following, inserting solutions (9) and (10) into (7) and (8) yields the following ODEs: We solve the systems (11)- (14); the solutions are shown as below: Then, the solutions of systems (5) and (6) can be shown by substituting the above functions ϕ 1 ðtÞ, ϕ 2 ðtÞ, ψ 1 ðtÞ, and ψ 2 ðtÞ into Eqs. (9) and (10).
admits the CLBS: The system (19) is a system of ODEs with respect to variable x, so the following forms are the corresponding solutions.
In the following, inserting solutions (21) into (16) yields the following ODEs:

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.