A geometric approach for solving the density-dependent diffusion Nagumo equation
- Mir Sajjad Hashemi^{1}Email author,
- Elham Darvishi^{2} and
- Dumitru Baleanu^{3, 4}
https://doi.org/10.1186/s13662-016-0818-2
© Hashemi et al. 2016
Received: 30 October 2015
Accepted: 22 March 2016
Published: 31 March 2016
Abstract
In this paper, some solutions of the density-dependent diffusion Nagumo equation are obtained by using a new approach, the Lie symmetry group-preserving scheme (LSGPS). The effects of various model parameters on the solution are investigated graphically using LSGPS. Finally, a different reduction method of PDEs is applied to construct two new analytical solutions and a first integral of the Nagumo equation.
Keywords
1 Introduction
A systematic and powerful method to derive the exact solutions of nonlinear differential equations is the Lie symmetry method which has some important properties such as conservation laws, can successfully be obtained using the symmetries [1–5]. Among the existing numerical algorithms, the group preserving scheme (GPS) provided by Liu [6] is a numerical method based upon Lie group solvers that preserves the Lie group structure under discretization. This method uses the Cayley transformation and the Padé approximations in the augmented Minkowski space \(\mathbb{M}^{n+1}\). One of the major benefits of GPS in the \(\mathbb{M}^{n+1}\) is that it can avoid ghost fixed points and spurious solutions. We refer the reader to the following papers about GPS, e.g. [7–16].
2 Group-preserving schemes
This section gives an introduction to GPS from the class of the relatively new area of numerical analysis called geometric integration. The name ‘geometric integration’ is utilized to a series of numerical approaches that aim to preserve the qualitative and geometrical features of a differential equation when it is discretized [23–25]. Hairer in [26] illustrated a procedure to modify classical methods for solving differential equations on manifolds in order to preserve certain geometric properties of the exact flow. The concept of geometric numerical integration by the important example of the Störmer/Verlet method is given by Hairer et al. in [27].
2.1 A Lie algebra formulation
2.2 Lie symmetry group preserving scheme
Suppose that the spatial (time) derivatives of (17) are discretized by some methods - finite differences, finite elements, finite volumes, or any other method. Then, this semi-discretization generates a system of ordinary differential equations that can be solved by GPS, but some problems may occur during this procedure. Semi-discretization is very sensitive to the choice of the discretization step size Δx (Δt), i.e. many criteria should be considered, e.g. consistency, stability, the Courant-Friedrichs-Levi (CFL) condition, and so on. Thus, if the choice of Δx (Δt) were either too little or too big then either the consistency or the stability of the discretization may be missed and thus the final solution of (17) could not be trusted. It may happen that Δx (Δt) has to be very small, e.g. \(\Delta x=0.0001\), in order to achieve a good result with semi-discretization, thus leading to a huge number of ODEs, e.g. \(\Delta x=10^{-n}\) means a system of n ODEs, for which the traditional GPS or Lie group shooting method will require a large number of operations and computations.
Instead the benefits of the LSGPS are several, mainly one needs just one ODE, while in the traditional GPS a large number of ODEs are necessary. Also since discretization reduces the accuracy and imposes the error, solutions obtained by LSGPS are more accurate than by GPS, because LSGPS uses the discretization in one dimension, while GPS uses the discretization in two dimensions.
3 Lie group analysis and LSGPS of density-dependent diffusion Nagumo equation
Reduction 1
Reduction 2
4 Final remarks
In this paper, we considered the density-dependent diffusion Nagumo equation within a new geometric method, LSGPS. Three dimensional Lie algebra and optimal system of Lie algebras related to equation (1) were obtained. After the reduction of this equation by utilizing the Lie symmetries, the traveling wave solutions are first discussed in Reduction 1. Another reduction of the Nagumo equation, by using Lie symmetries and Nucci’s method (Reduction 2), leads to a first integral and two new analytical solutions of equation (1).
\(\mathcal{A}\) is an element of the Lie algebra \(\mathit{so}(k,1)\) of the proper orthochronous Lorentz group \(\mathit{SO}_{0}(k, 1)\).
Declarations
Acknowledgements
The authors would like to thank the referees for their useful comments and remarks.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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