Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 6143934, 18 pages

https://doi.org/10.1155/2018/6143934

## Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

^{1}Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan^{2}Department of Mathematics, National College of Business Administration and Economics, Lahore 54000, Pakistan

Correspondence should be addressed to Muhammad Abbas

Received 17 August 2017; Revised 5 November 2017; Accepted 13 December 2017; Published 18 January 2018

Academic Editor: Chaudry M. Khalique

Copyright © 2018 Imtiaz Wasim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

#### 1. Introduction

Nonlinear partial differential equations (PDEs) play a significant role in different fields of science and engineering. Many physical problems are designed in mathematical form using nonlinear PDEs [1–3]. The generalized Burgers-Fisher (GBF) and generalized Burgers-Huxley (GBH) equations describe many physical phenomena. Numerical treatment of these two equations have become a dominant tool due to complexities in finding their solutions.

Consider one-dimensional nonlinear partial differential equation of the following form:subject to the initial condition,and the boundary conditions,where and are some nonlinear expressions in terms of provided that for GBF equation and for GBH equation while , , , , are constants such that , , , . The nonlinear diffusion models generated from (1) have a significant role in nonlinear physics and of great practical interest.

##### 1.1. Model I

The GBF equation has a lot of applications in the fields such as fluid mechanics [4], gas dynamics, plasma physics [4], number theory, elasticity [5], and heat conduction [6]. This equation becomes the classical Fisher equation when , which is one of the significant structures in population biology and is given byKolmogorov et al. [7] wrote down the same equation for the description of dynamic spread of a combustion front. It arises in several phenomena involving perturbation spreads in excitable mediums, spreading of bacterial colonies [8], spread of reaction fronts in chemically bistable systems [9], and switching in nonlinear optics [10].

The exact solution of GBF issubject to the initial condition,and the boundary conditions,whereWang et al. [11] investigated the exact solution of GBH equation with the help of nonlinear transformations.

##### 1.2. Model II

Satsuma et al. [12] investigated the GBH equation in 1987. This equation reduces to the Huxley equation [11] when , , which describes nerve pulse propagation in nerve fibres and wall motion in liquid crystals [13, 14]. It can be expressed as follows:By considering a well-known experiment in liquid crystals, a similarity between the motion of a wall in liquid crystals and nerve propagation was discussed in [14]. These models have been studied widely in the last decades due to their importance in neurobiology. Hodgkin and Huxley [15] suggested their famous Hodgkin-Huxley model for nerve propagation in 1952. It takes the form of Burgers equation by considering , , . In nonlinear dissipative systems [16], it describes the far field of wave propagation and can be expressed as follows:It becomes a FitzHugh-Nagumo (FN) equation when , , , are chosen. Basically, it is reaction diffusion equation utilized in circuit theory and biology [17] and its mathematical form isWhen , , , this equation also reduces to prototype model named as Burgers-Huxley equation. It describes the interaction between diffusion transports, convection, and reaction mechanisms [18] and is given byThe exact solution of GBH equation can be written as follows:subject to the initial condition,and the boundary conditions,where This solution was investigated by Xinyi and Yuekai [19] which is the generalization of the preceding results.

Several numerical techniques have been developed to find the numerical solution of GBF and GBH equations. Javidi presented the numerical solution of GBH equation using spectral collocation method [20] and pseudospectral and preconditioning [21] and Chebyshev polynomials to develop a new domain decomposition algorithm [22]. Golbabai and Javidi [23] applied a spectral domain decomposition technique for the numerical solution of GBF equation. Darvishi et al. [24] investigated the numerical solution of GBH equation by adopting a spectral collocation method and Darvishi et al.’s preconditioning. Sari et al. [25] presented the numerical solution of GBF equation by applying a compact finite difference scheme. Hammad and El-Azab [26] computed the numerical solution of two types of equations, namely, GBF and GBH, using 2 order compact finite difference scheme. A computational meshless method was developed by Khattak [27] for solving the GBH equation.

Sari and Gurarslan [28] obtained the numerical solution of the GBH equation using a polynomial differential quadrature method. Malik et al. [29] developed a heuristic scheme for the numerical solution of the GBF equation based on the hybridization of Exp-function method with nature inspired algorithm. The problem was converted into a nonlinear ordinary differential equation (ODE) by substitution. The travelling wave solution was approximated by the Exp-function method with unknown parameters. Dehghan et al. [30] developed two numerical methods based on the interpolating scaling functions and mixed collocation finite difference schemes for the numerical solution of the GBH equation.

Zhang et al. [31] developed a new kind of exact finite difference scheme for solving Burgers equation and Burgers-Fisher equation using the solitary wave solution. Biazar and Muhammadi [32] solved GBH equation using differential transform method (DTM). Bratsos [33, 34] solved GBH equation using a modified predictor-corrector method based on a second- and fourth-order time finite difference scheme. Zarebnia and Aliniya [35] used a mesh-free collocation method based on sinc functions for solving the Burgers-Huxley equation. Batiha et al. [36] applied He’s variational iteration method (VIM) without any discretization to solve the GBH equation. Morufu [37] developed an improved algorithm for solving GBF equation based on a Maple code. Hashim et al. [38] applied Adomian decomposition method (ADM) to get rapidly convergent analytical series solution of the GBH equation.

Zhao et al. [39] approximated the GBF equation using the pseudospectral method based on Crank-Nicolson/leapfrog scheme. The approximate solutions were obtained for the GBH and GBF equations using the Adomian and discrete Adomian decomposition methods [40, 41]. Inan and Bahadir [42] obtained a numerical solution of the GBH equation using implicit exponential finite difference method. Celik [43] proposed a Chebyshev wavelet collocation method based on truncated Chebyshev wavelet series for the solution of GBH equation. Moreover, the numerical solution of proposed GBH equation was obtained using several numerical methods named Galerkin method [44], implicit and fully implicit exponential finite difference methods [45], Haar wavelet method [46], conditionally bounded and symmetry-preserving method [47], linearly implicit compact scheme [48], positive and bounded finite element method [49], explicit solution scheme [50], exponential time differencing scheme [51], and higher order finite difference schemes [52].

The B-spline collocation scheme is a well-known interpolating or approximating scheme which provides a good approximation rate, computationally fast, numerically consistent, and has ability to reproduce the shape of the data with second order of continuity as compared to polynomials. Recently, several numerical schemes based on different types of B-spline functions were applied to find the numerical solutions of the differential equations. Mittal and Tripathi [53] proposed a numerical scheme based on modified cubic B-spline functions to get the approximate solutions of GBF and GBH equations. Mittal and Jain [54] obtained a numerical solution of nonlinear Burgers equation using a modified cubic B-spline collocation method. Singh et al. [55] developed a numerical scheme for solving the GBH equation using modified cubic B-spline differential quadrature method (MCB-DQM) and numerical results can be obtained using SSP-RK43 scheme. Reza [56] implemented the cubic B-spline collocation scheme based on the finite difference scheme for solving the GBH equation. Reza [57] developed a numerical method based on exponential B-spline with finite difference approximations to solve the GBF equation. Recently, Bukhari et al. [58] applied local radial basis functions differential collocation (LRBDQ) method to compute the numerical solution of GBH equation.

##### 1.3. Motivation of the Study

The finite difference scheme is not the only tool for computing approximations to the solution of boundary value problems. There are various approximation techniques which have been examined by many researchers. Spline interpolation method is one of the most effective approximation methods on account of its simplicity and practicality. The main advantage of using this method is that it is able to approximate the analytical curve up to certain smoothness. Therefore, the spline method has the flexibility to get the approximation at any point in the domain with more accurate results compared to the usual finite difference method. This, thus, provides the motivation for this study on examining the accuracy of hybrid B-spline on solving nonlinear partial differential equation. However, one of the limitations of classical B-spline interpolation is that it does not possess any free parameter for the curve modification. Therefore, the shape of the curve is incapable of being altered once the control points are determined. On the other hand, spline interpolation is a global interpolation; any changes of the data point will require solving all the linear systems again. The advantage of using hybrid B-spline is that it possesses a free parameter to control the global shape of curve. An appropriate choice of the parameter rises the order of accuracy of the scheme. Hybrid B-spline basis function reduces to cubic trigonometric B-spline and cubic B-spline function when and , respectively. This research focuses on the value of . Figure 1 depicts the graph of cubic trigonometric B-spline when , cubic B-spline function when , and the effect of parameter for proposed hybrid B-spline function. Therefore, the superiority of this spline interpolation method on proposed problem is to be examined.