# Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions

- Anjali Verma
^{1}, - Ram Jiwari
^{2}Email author and - Mehmet Emir Koksal
^{3}

**2014**:229

https://doi.org/10.1186/1687-1847-2014-229

© Verma et al.; licensee Springer. 2014

**Received: **3 May 2014

**Accepted: **5 August 2014

**Published: **20 August 2014

## Abstract

In this article, the authors study analytic and numerical solutions of nonlinear diffusion equations of Fisher’s type with the help of classical Lie symmetry method. Lie symmetries are used to reduce the equations into ordinary differential equations (ODEs). Lie group classification with respect to time dependent coefficient and optimal system of one-dimensional sub-algebras is obtained. Then sub-algebras are used to construct symmetry reduction and analytic solutions. Finally, numerical solutions of nonlinear diffusion equations are obtained by using one of the differential quadrature methods.

## Keywords

## 1 Introduction

in which $\alpha >0$ is a parameter. Kawahara and Tanaka [2] found an exact solution describing the coalescence of two traveling wave fronts of (2). Applications of traveling wave fronts appear in biology, chemistry, and medicine [3]. Such wave fronts were studied by Fisher for the first time in 1930s by considering (2).

The Fisher equation (2) appears in chemical kinetics [4], in logistic population growth models [5], autocatalytic chemical reactions, branching Brownian motion processes, flame propagation, and neurophysiology. The reaction-diffusion equation (2) also expresses a model equation for the evolution of a neutron population in a nuclear reactor [6] and also arises in the study of chemical wave propagation [7]. This equation includes the effects of linear diffusion via ${u}_{xx}$ and nonlinear local multiplication or reaction via $u(1-u)$.

There is a large cycle of works on mathematical properties and discussion of the Fisher equation in the literature. Larson [8], Kawahara and Tanaka [2], and Brazhnik and Tyson [5] provided excellent summaries of the Fisher equation. Ablowitz and Zepetella [9] gave the presentation of explicit solutions of the Fisher equation for a special wave speed. For the generalized the Fisher equation, Wang [10] presented the exact and explicit solitary wave solutions. Then Wazwaz and Gorguis [11] considered an analytic study of the Fisher equation by using the Adomian decomposition method. But it was not before 1974 when numerical solutions of the Fisher equation were available in the literature. Gazdag and Canosa [12] were the first to study numerical solutions of the Fisher equation with a pseudo-spectral approach. Afterwards, a lot of researchers have studied numerical solutions on the Fisher equation. Hagstrom and Keller [13] presented asymptotic boundary conditions by using a centered finite-difference algorithm. Afterwards, Evans and Sahimi [14] used an alternating group explicit iterative method to solve (2) and obtained satisfactory results, of a qualitatively similar nature. The numerical scheme considered in [14] is quite complicated and it causes unexpected high-frequency oscillations, which must be filtered out at each time step. Next, Parekh and Puri [15] and Twizell *et al*. [16] developed implicit and explicit finite differences algorithms for numerically solving the Fisher equation. Then Tang and Weber [17] proposed a Galerkin finite element method. Mickens [18] put forward a best finite-difference scheme for the Fisher equation. Afterwards, Garey and Shen [19] used a least-squares finite element method and Qiu and Sloan [20] used a moving mesh method for numerical solution of the Fisher equation. Rizwan [21] compared the nodal integral method and non-standard finite-difference schemes, and Khaled [22] proposed the Sinc collocation method. Daniel *et al.* [23] proposed a pseudo-spectral method for the numerical solution of the Fisher equation. Mittal and Kumar [24] studied the Fisher equation by applying wavelet Galerkin method, while Jiwari *et al*. [25–30] studied numerical solutions of some nonlinear evolution equations by using differential quadrature method. Finally, for numerical solutions of the nonlinear Fisher reaction-diffusion equation, Mittal and Jain [31] proposed a numerical method, based on collocation of modified cubic *B*-splines over finite elements.

The purpose of this paper is to present analytic and numerical solutions via symmetry reductions of nonlinear diffusion equations of Fisher’s type defined as:

in which *a* is a parameter.

Lie symmetries are used to reduce the equations to ordinary differential equations (ODEs). In the next section, lie group classification with respect to time dependent coefficient and optimal system of one-dimensional sub-algebras is obtained. Then, in Section 3, sub-algebras are used to construct symmetry reduction and analytic solutions. In Section 4, numerical solutions of nonlinear diffusion equations are obtained by using polynomial differential quadrature method. Finally, in Section 5, absolute, root mean square (RMS), and ${L}_{\mathrm{\infty}}$ errors are calculated.

## 2 Method of Lie symmetries

*n*in

*p*-independent and

*q*-dependent variables is given as a system of equations,

*u*with respect to

*x*up to

*n*, where ${u}^{(n)}$ expresses all the derivatives of

*u*of all orders from 0 to

*n*. We consider a one-parameter Lie group of infinitesimal transformation acting on the independent and dependent variables of the system (6)

*s*is the parameter of transformation and ${\xi}^{i}$, ${\eta}^{j}$ are infinitesimals of transformation for the independent and dependent variables, respectively. The invariance of the system (6) under the infinitesimal transformation leads to the invariance conditions

*n*th-order prolongation of infinitesimal generator given by

*j*. If $j=k$ the coefficients ${\varphi}_{j}^{\alpha}$ of $\partial {u}_{j}^{\alpha}$ will only depend on

*k*th- and lower-order derivatives of

*u*;

where ${u}_{i}^{\alpha}=\frac{\partial {u}^{\alpha}}{\partial {x}^{i}}$ and ${u}_{j,i}^{\alpha}=\frac{\partial {u}_{j}^{\alpha}}{\partial {x}^{i}}$.

One of the most important properties of these infinitesimal symmetries is that they form a Lie algebra under the usual Lie bracket.

## 3 Polynomial differential quadrature method

where ${a}_{ij}$ and ${w}_{ij}^{(r)}$ are the weighting coefficients of the first-order derivative and *r* th-order derivative, respectively.

## 4 Lie classical analysis for nonlinear diffusion Fisher’s type equations

*ε*is the group parameter. The functions

*ξ*,

*τ*,

*η*are the infinitesimals of the transformations for the variables

*x*,

*t*, and

*u*, respectively. We shall denote the infinitesimals for ${u}_{t}$, ${u}_{xx}$ by ${\eta}^{x}$, ${\eta}^{xx}$. The infinitesimals are as follows:

*V*is given by

Now, substitute (23) and (25) into (24). Then we collect together the coefficients of *u*, ${u}_{x}$, ${u}_{t}$, ${u}_{xx}$, ${u}_{tt}$ and set all of them to zero. Finally, we get a system of linear partial differential equations from which we can find *ξ*, *τ*, and *η* in practice.

*u*, ${u}_{x}$, ${u}_{t}$, ${u}_{xx}$ to zero. Then we obtain

*a*and

*b*are arbitrary constants. The associated vector fields for the one-parameter Lie group of infinitesimal transformations are ${V}_{1}$ and ${V}_{2}$ as given by

Our task is to simplify as many of the coefficients ${\epsilon}_{i}$, $i=1,2$ as possible though judicious applications of adjoint maps to *V*. By taking ${a}_{2}\ne 0$, ${a}_{2}=1$, we have ${V}_{2}+{a}_{1}{V}_{1}$.

### 4.1 Reductions by one-dimensional sub-algebras

where ${c}_{1}$ is a constant.

In this similar way, we can apply the Lie classical method to (4) and (5) and can study the exact solutions of these equations.

where ${c}_{2}$ is a constant.

Corresponding to the basic vector field ${V}_{1}$ in the optimal system, we can obtain only a constant solution.

where ${c}_{3}$ is an arbitrary constant.

For the sub-algebra ${V}_{1}$, only a constant can be obtained.

## 5 Comparative study of exact and numerical solutions of Fisher’s type equations

where *α*, *a* are parameters, ${b}_{ij}$ are weighting coefficients of the second-order partial derivative and ${u}_{i}=u({x}_{i},t)$. Equations (39)-(40) are systems of first-order nonlinear differential equations. The initial and boundary conditions are taken from the analytic solutions obtained by Lie symmetry method in Section 4. Finally, the systems of initial and boundary value problems are solved by Pike and Roe’s fourth-stage RK4 [34].

### 5.1 Numerical experiments and discussion

where ${e}_{i}=({u}_{i}-{U}_{i})$, ${u}_{i}$ are approximated solutions and ${U}_{i}$ are exact solutions.

**Example 1**Consider (3) over the domain $[0,1]$ with the following initial conditions:

where $c=\frac{-5\sqrt{\alpha}}{\sqrt{6}}$ and ${c}_{1}$ is an arbitrary constant.

${\mathit{L}}_{\mathbf{\infty}}$
**, RMS, and**
${\mathit{L}}_{\mathbf{2}}$
**errors of Example**
**1**
**at different times**
t
**and for different constants**

t | ${\mathit{c}}_{\mathbf{3}}\mathbf{=}\mathbf{1}$ | ${\mathit{c}}_{\mathbf{3}}\mathbf{=}\mathbf{3}$ | ||||
---|---|---|---|---|---|---|

${\mathit{L}}_{\mathbf{\infty}}$ | RMS | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS | ${\mathit{L}}_{\mathbf{2}}$ | |

0.2 | 5.659E−06 | 6.500E−06 | 1.578E−05 | 1.362E−07 | 3.685E−08 | 1.801E−05 |

0.5 | 6.306E−06 | 7.278E−06 | 1.547E−05 | 1.581E−07 | 4.278E−08 | 1.800E−05 |

1.0 | 7.457E−06 | 8.691E−06 | 1.489E−05 | 2.026E−07 | 5.485E−07 | 1.798E−05 |

3.0 | 1.175E−05 | 1.471E−05 | 1.165E−05 | 5.432E−07 | 1.472E−07 | 1.786E−05 |

5.0 | 1.175E−05 | 1.705E−05 | 7.666E−06 | 1.422E−06 | 3.870E−07 | 1.753E−05 |

**Example 2**In this example, we have considered the Fisher equation (4) over the domain $[0,1]$ with the following initial conditions:

${\mathit{L}}_{\mathbf{\infty}}$
**, RMS, and**
${\mathit{L}}_{\mathbf{2}}$
**errors of Example**
**2**
**at different times**
t
**and for different constants**

t | α = 1, ${\mathit{c}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$ | α = 2, ${\mathit{c}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$ | ||||
---|---|---|---|---|---|---|

${\mathit{L}}_{\mathbf{\infty}}$ | RMS | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS | ${\mathit{L}}_{\mathbf{2}}$ | |

0.2 | 2.105E−05 | 3.633E−03 | 4.475E−03 | 4.217E−05 | 1.903E−03 | 2.300E−03 |

0.5 | 1.736E−05 | 3.821E−03 | 4.444E−03 | 2.699E−05 | 1.401E−03 | 1.548E−03 |

1.0 | 1.103E−05 | 2.464E−03 | 2.690E−03 | 9.083E−06 | 4.226E−03 | 4.383E−03 |

3.0 | 3.788E−05 | 2.245E−05 | 2.277E−05 | 8.711E−05 | 9.755E−05 | 9.768E−05 |

5.0 | 2.777E−07 | 1.818E−07 | 1.823E−07 | 5.871E−06 | 3.486E−06 | 3.486E−06 |

**Example 3**Consider the Fisher equation (5) over the domain $[0,1]$ with the following initial and boundary conditions:

${\mathit{L}}_{\mathbf{\infty}}$
**, RMS, and**
${\mathit{L}}_{\mathbf{2}}$
**errors of Example**
**3**
**at different times**
t
**and for different constants**

t | a = 0.5, ${\mathit{c}}_{\mathbf{3}}\mathbf{=}\mathbf{1}$ | a = 1.5, ${\mathit{c}}_{\mathbf{3}}\mathbf{=}\mathbf{1}$ | ||||
---|---|---|---|---|---|---|

${\mathit{L}}_{\mathbf{\infty}}$ | RMS | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS | ${\mathit{L}}_{\mathbf{2}}$ | |

0.2 | 2.412E−06 | 3.045E−06 | 7.903E−07 | 2.105E−05 | 5.726E−03 | 7.054E−03 |

0.5 | 2.227E−06 | 2.802E−06 | 7.225E−07 | 1.736E−05 | 5.340E−03 | 6.210E−03 |

1.0 | 1.936E−06 | 2.425E−06 | 6.193E−07 | 1.103E−05 | 3.055E−03 | 3.335E−03 |

3.0 | 1.039E−06 | 1.285E−06 | 3.196E−07 | 6.243E−05 | 3.717E−05 | 3.767E−05 |

5.0 | 5.228E−07 | 6.425E−07 | 1.577E−07 | 6.174E−07 | 3.750E−07 | 3.760E−07 |

## 6 Conclusion

In this article, the authors studied analytic and numerical solutions of the Fisher type equations with the help of the classical Lie symmetry method and the polynomial differential quadrature method. The Lie symmetry method is utilized to investigate the symmetries and invariant solutions of the equations. By determining the transformation group under which a given system is invariant, information about the invariants and symmetries of that equation is obtained. This information, in turn, is used to determine similarity variables that reduce the number of independent variables. The vector fields of the optimal system lead to a reduction of the nonlinear system of partial differential equations to ordinary differential equations. The infinitesimal generators in the optimal system are used for reductions and exact solutions. Finally, the polynomial differential quadrature method is used to find the numerical solutions of the Fisher type equations with the help of initial and boundary conditions taken from the analytic solutions obtained by the classical Lie symmetry method. It is concluded that the numerical solutions are in good agreement with the analytical solutions. ${L}_{\mathrm{\infty}}$, RMS, and ${L}_{2}$ errors are calculated for each equation with particular values of arbitrary constants, which are small and negligible.

## Declarations

### Acknowledgements

The authors are very thankful to the reviewers for their valuable suggestions to improve the quality of the paper.

## Authors’ Affiliations

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