A fourthorder Bspline collocation method for nonlinear Burgers–Fisher equation
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Abstract
A fourthorder Bspline collocation method has been applied for numerical study of Burgers–Fisher equation, which illustrates many situations occurring in various fields of science and engineering including nonlinear optics, gas dynamics, chemical physics, heat conduction, and so on. The present method is successfully applied to solve the Burgers–Fisher equation taking into consideration various parametric values. The scheme is found to be convergent. Crank–Nicolson scheme has been employed for the discretization. Quasilinearization technique has been employed to deal with the nonlinearity of equations. The stability of the method has been discussed using Fourier series analysis (von Neumann method), and it has been observed that the method is unconditionally stable. In order to demonstrate the effectiveness of the scheme, numerical experiments have been performed on various examples. The solutions obtained are compared with results available in the literature, which shows that the proposed scheme is satisfactorily accurate and suitable for solving such problems with minimal computational efforts.
Keywords
Burgers–Fisher equation Cubic Bspline Collocation method Crank–Nicolson method Gauss elimination methodIntroduction
Recently, various numerical and analytical methods have been used by various researchers to deal with the Burgers–Fisher equation. In 2004, Kaya and ElSayed numerically simulated the generalized Burgers–Fisher equation [23] and came up with its explicit solutions. Ismail et al. [24] applied Adomian decomposition method (ADM), Javidi [25] employed modified pseudospectral method, Rashidi et al. [26] used homotopy perturbation method (HPM), Khattak [27] employed collocationbased radial basis functions method (CBRBF) and Xu and Xian [28] applied Expfunction method to find the analytic as well as numerical solutions of the generalized Burgers–Fisher equation. Also many other authors used different methods to obtain the analytical and numerical solution of the generalized Burgers–Fisher equation; for example, Zhu and Kang [29] used the Bspline quasiinterpolation method, Zhang and Yan [30] used a lattice Boltzmann model, Sari et al. [31] used the compact finite difference method, Sari et al. [32] developed the polynomialbased differential quadrature method, Zhang et al. [33] used the local discontinuous Galerkin (LDG) methods and Nawaz et al. [34] employed optimal homotopy asymptotic method (OHAM).
Very recently, Yadav and Jiwari [35] employed Galerkin’s finite element method to analyze and approximate the Burgers–Fisher equation. S Malik, Qureshi, Amir, A Malik and Haq [36] used the Expfunction method hybridized with heuristic computation for the numerical simulation of the Burgers–Fisher equation. In 2015, Mittal and Tripathi developed a collocation method using cubic Bsplines to numerically solve generalized Burgers–Fisher and generalized Burgers–Huxley equations [37].
Recently, Bspline functions have gained popularity as a powerful tool in the field of image processing, approximation theory and numerical simulation of boundary and initial value problems. Bsplines as basis functions have been used in various numerical methods such as Bspline differential quadrature method and Bspline collocation method to deal with the partial differential equations. Cubic Bspline collocation method is used by Goh et al. [38] to solve heat and advection diffusion equations in one dimension. Dag and Saka [39] used the Bspline collocation method for equalwidth equation. Bspline collocation method has been also used by Kadalbajoo and Arora [40] to deal with the singular perturbation problems and by Zahra [41] to study PHIfour and Allen–Cahn equations. Ersoy and Dag [42] applied this method to solve Kuramoto–Sivashinsky equation. Khater et al. [43] obtained numerical solution of the Burgerstype equations by using cubic spline collocation method.
In the proposed work, the fourthorder cubic Bspline collocation method is adopted to solve Burgers–Fisher equation. Fourthorder approximation for both single and double derivatives is employed. It has been done by using different end conditions and taking one more term in the Taylor series expansion, thus resulting in very accurate and efficient numerical solutions. Moreover, the present method does not require any involvement of integrals to get the final set of equations, thus reducing the computational efforts to a great extent.
The aim of this work is to investigate the numerical solutions of the Burgers–Fisher equation for different parametric values using collocation method with cubic Bsplines as basis functions.
To the best of our knowledge, nobody has yet dealt with the Burgers–Fisher equation with the scheme considered in this work. The present scheme gives the approximate solution at any point of the solution domain. Our work is compared with the previous literature, and results are found to be better in terms of accuracy and efficiency. The proposed method is quite simple and produces highly accurate results for considerably lesser grid size, hence reducing complexity and computational cost.
The organization of this paper is as follows. “Mathematical formulation” section gives a description of the cubic Bspline collocation method. In “Implementation of the method” section, the method is applied to the Burgers–Fisher equation with the treatment of boundary conditions. In “Stability of the scheme” section, stability analysis of the method is carried out. “Numerical experiments and discussions” section presents some test examples of the Burgers–Fisher equation. A summary is given at the end of the paper in “Conclusion” section.
Mathematical formulation
The approximate values K(x, t) and their firstorder derivatives at the knots are defined using Taylor expansions and finite difference as follows:
Implementation of the method
Stability of the scheme
Hence, the proposed collocation method using Bsplines as basis function is unconditionally stable.
Numerical experiments and discussions
Example 1
Absolute error comparison of the present method with FEM [35] for \(\alpha =1\) and \(\beta =1\)
Absolute errors at grid points for \(\alpha =1, \beta =1\) at different times T
x  T = 0.001  T = 0.005  T = 0.01 

0.1  1.7078E−008  2.1434E−007  3.2389E−007 
0.2  5.0626E−012  2.9936E−008  1.0350E−007 
0.3  1.1102E−014  1.7287E−009  2.1711E−008 
0.4  1.1657E−014  3.9512E−011  2.9320E−009 
0.5  2.5313E−014  9.0339E−013  5.5391E−010 
0.6  3.4306E−014  4.6625E−011  3.4449E−009 
0.7  3.4528E−014  1.9248E−009  2.4153E−008 
0.8  5.3543E−012  3.1502E−008  1.0880E−007 
0.9  1.7003E−008  2.1318E−007  3.2173E−007 
Example 2
Absolute error comparison of the present scheme with different schemes for \(\alpha =0.001\) and \(\beta =0.001\)
x  T  Present method  FEM [35]  EFM [36]  OHAM [34]  CFDM [31] 

0.1  0.001  4.64E−011  1.21E−009  1.97E−008  2.25E−008  1.01E−007 
0.005  5.85E−010  1.69E−009  1.97E−008  1.12E−007  4.38E−007  
0.01  8.84E−010  1.28E−009  1.97E−008  2.25E−007  7.53E−007  
0.5  0.001  5.39E−014  2.28E−012  3.58E−009  4.58E−008  1.04E−007 
0.005  2.73E−013  2.49E−009  3.71E−009  2.29E−007  5.21E−007  
0.01  2.04E−012  2.50E−009  3.88E−009  4.58E−007  1.04E−006  
0.9  0.001  4.64E−011  1.20E−010  1.80E−008  4.58E−008  1.01E−007 
0.005  5.85E−010  1.69E−009  1.77E−008  2.29E−007  4.38E−007  
0.01  8.84E−010  1.28E−009  1.74E−008  4.58E−007  7.53E−007 
Absolute errors at grid points for \(\alpha =0.001, \beta =0.001\) at different times T
x  T = 0.001  T = 0.005  T = 0.01 

0.1  4.6470E−011  5.8512E−010  8.8446E−010 
0.2  7.0222E−014  8.4073E−011  2.9048E−010 
0.3  5.5789E−014  5.2426E−012  6.3005E−011 
0.4  5.6677E−014  3.9008E−013  9.1993E−012 
0.5  5.3957E−014  2.7317E−013  2.0401E−012 
0.6  5.4845E−014  3.8886E−013  9.1954E−012 
0.7  5.4456E−014  5.2395E−012  6.3002E−011 
0.8  7.0832E−014  8.4079E−011  2.9048E−010 
0.9  4.6478E−011  5.8515E−010  8.8450E−010 
CPUtime (s)  12.7352  63.0485  125.7683 
Comparison of absolute errors with EFM [36] for time T = 0.1 and \(\alpha =0.001, \beta =0.001\)
x  Present method  EFM [36] 

0.1  1.575E−009  1.988E−008 
0.2  1.332E−009  1.706E−008 
0.3  1.139E−009  1.390E−008 
0.4  1.015E−009  1.040E−008 
0.5  9.731E−010  6.547E−009 
0.6  1.015E−009  2.354E−009 
0.7  1.139E−009  2.182E−009 
0.8  1.332E−009  7.062E−009 
0.9  1.575E−009  1.228E−008 
Example 3
Comparison of numerical solutions of the present method with analytic solutions of HPM [26] at a different set of values of \(\alpha \) and \(\beta \)
T  x  \(\alpha =0.5, \beta =0.5\)  \(\alpha =0.1, \beta =0.1\)  

Present method  HPM [26]  Present method  HPM [26]  
0.1  0.2  1.2205E−006  6.1768E−006  2.1998E−007  4.3262E−008 
0.4  9.2477E−007  1.6029E−005  1.6726E−007  1.0883E−007  
0.6  9.0491E−007  2.5802E−005  1.666E−007  1.7457E−007  
0.8  1.1797E−006  3.5447E−005  2.1871E−007  2.4012E−007  
0.4  0.2  1.6342E−006  7.8774E−005  2.9985E−007  3.8516E−007 
0.4  1.6248E−006  7.8951E−005  2.9715E−007  6.6533E−007  
0.6  1.6277E−006  2.3628E−004  2.9713E−007  1.7158E−006  
0.8  1.6438E−006  3.9244E−004  2.9982E−007  2.7658E−006  
0.8  0.2  1.6046E−006  1.2446E−003  3.0381E−007  7.2803E−006 
0.4  1.6189E−006  6.2245E−004  3.0384E−007  3.0801E−006  
0.6  1.6316E−006  2.8091E−006  3.0392E−007  1.1209E−006  
0.8  1.6427E−006  6.2804E−004  3.0403E−007  5.3215E−006 
Example 4
Comparison of \(E_\mathrm{A}\) and \(E_\mathrm{R}\) of the present method with PDQM [32] at different times T for different parametric values
x  T  \(\alpha =0.01, \beta =0.01\)  \(\alpha =0.0001, \beta =0.0001\)  

Present method  PDQM [32]  Present method  PDQM [32]  
\(E_\mathrm{A}\)  \(E_\mathrm{R}\)  \(E_\mathrm{A}\)  \(E_\mathrm{R}\)  \(E_\mathrm{A}\)  \(E_\mathrm{R}\)  \(E_\mathrm{A}\)  \(E_\mathrm{R}\)  
\(x_{3}\)  1  7.91E\(\)007  1.57E\(\)006  2.14E\(\)005  4.27E\(\)005  7.97E\(\)009  1.57E\(\)008  2.15E\(\)007  4.31E\(\)007 
10  7.89E\(\)007  1.50E\(\)006  2.04E\(\)005  3.89E\(\)005  7.97E\(\)009  1.57E\(\)008  2.15E\(\)007  4.30E\(\)007  
50  7.43E\(\)007  1.19E\(\)006  1.53E\(\)005  2.46E\(\)005  7.96E\(\)009  1.57E\(\)008  2.15E\(\)007  4.29E\(\)007  
\(x_{8}\)  1  7.98E\(\)007  1.59E\(\)006  1.28E\(\)004  2.56E\(\)004  7.96E\(\)009  1.59E\(\)008  1.29E\(\)006  2.58E\(\)006 
10  7.96E\(\)007  1.51E\(\)006  1.22E\(\)004  2.33E\(\)004  7.95E\(\)009  1.59E\(\)008  1.29E\(\)006  2.57E\(\)006  
50  7.50E\(\)007  1.20E\(\)006  9.16E\(\)005  1.47E\(\)004  7.94E\(\)009  1.58E\(\)008  1.28E\(\)006  2.56E\(\)006  
\(x_{13}\)  1  7.98E\(\)007  1.59E\(\)006  4.49E\(\)005  8.95E\(\)005  7.97E\(\)009  1.59E\(\)008  4.50E\(\)007  9.00E\(\)007 
10  7.96E\(\)007  1.51E\(\)006  4.28E\(\)005  8.16E\(\)005  7.98E\(\)009  1.59E\(\)008  4.50E\(\)007  9.00E\(\)007  
50  7.50E\(\)007  1.20E\(\)006  3.20E\(\)005  5.15E\(\)005  7.98E\(\)009  1.58E\(\)008  4.49E\(\)007  8.95E\(\)007 
Absolute errors at grid points for \(\alpha =0.0001, \beta =0.0001\) at different times T
x  \( T=1 \)  \( T=10 \)  \( T=50 \) 

\(x_{1}\)  8.6414E\(\)009  8.6415E\(\)009  8.6415E\(\)009 
\(x_{2}\)  7.8946E\(\)009  7.8948E\(\)009  7.8948E\(\)009 
\(x_{3}\)  7.9700E\(\)009  7.9703E\(\)009  7.9702E\(\)009 
\(x_{4}\)  7.9623E\(\)009  7.9626E\(\)009  7.9626E\(\)009 
\(x_{5}\)  7.9630E\(\)009  7.9634E\(\)009  7.9634E\(\)009 
\(x_{6}\)  7.9628E\(\)009  7.9633E\(\)009  7.9633E\(\)009 
\(x_{7}\)  7.9628E\(\)009  7.9633E\(\)009  7.9633E\(\)009 
\(x_{8}\)  7.9628E\(\)009  7.9633E\(\)009  7.9633E\(\)009 
\(x_{9}\)  7.9628E\(\)009  7.9633E\(\)009  7.9633E\(\)009 
\(x_{10}\)  7.9628E\(\)009  7.9633E\(\)009  7.9633E\(\)009 
\(x_{11}\)  7.9630E\(\)009  7.9634E\(\)009  7.9634E\(\)009 
\(x_{12}\)  7.9623E\(\)009  7.9626E\(\)009  7.9626E\(\)009 
\(x_{13}\)  7.9700E\(\)009  7.9703E\(\)009  7.9702E\(\)009 
\(x_{14}\)  7.8946E\(\)009  7.8948E\(\)009  7.8948E\(\)009 
\(x_{15}\)  8.6414E\(\)009  8.6415E\(\)009  8.6415E\(\)009 
CPUtime(s)  5.2598  50.4549  250.6945 
Example 5
Comparison of \(E_\mathrm{A}\) and \(E_\mathrm{R}\) of the present method with PDQM [32]
x  T  \(\alpha =0.1, \beta =0.1\)  \(\alpha =0.01, \beta =0.01\)  

Present method  PDQM [29]  Present method  PDQM [29]  
\(E_\mathrm{A}\)  \(E_\mathrm{R}\)  \(E_\mathrm{A}\)  \(E_\mathrm{R}\)  \(E_\mathrm{A}\)  \(E_\mathrm{R}\)  \(E_\mathrm{A}\)  \(E_\mathrm{R}\)  
\(x_{3}\)  0.01  2.97E−006  5.92E−006  4.18E−005  8.35E−005  2.91E−007  5.83E−007  6.89E−006  1.38E−005 
0.1  6.59E−006  1.30E−005  1.47E−004  2.92E−004  6.46E−007  1.29E−006  2.84E−005  5.68E−005  
1  8.04E−006  1.52E−005  2.05E−004  3.89E−004  7.91E−007  1.57E−006  7.97E−005  1.59E−004  
\(x_{8}\)  0.01  1.55E−008  3.07E−008  1.03E−004  2.04E−004  1.53E−009  3.06E−009  1.04E−005  2.07E−005 
0.1  4.35E−006  8.56E−006  7.83E−004  1.54E−003  4.26E−007  8.51E−007  1.02E−004  2.04E−004  
1  8.13E−006  1.53E−005  1.21E−003  2.28E−003  7.98E−007  1.58E−006  4.68E−004  9.29E−004  
\(x_{13}\)  0.01  6.18E−007  1.21E−006  7.14E−005  1.40E−004  6.00E−008  1.19E−007  9.68E−006  1.93E−005 
0.1  5.43E−006  1.06E−005  2.94E−004  5.72E−004  5.30E−007  1.05E−006  5.39E−005  1.07E−004  
1  8.12E−006  1.51E−005  4.18E−004  7.78E−004  7.98E−007  1.58E−006  1.65E−004  3.27E−004 
Conclusion

The fourthorder cubic Bspline method has been adopted to numerically solve nonlinear Burgers–Fisher equation.

Crank–Nicholson for discretization and quasilinearization to deal with the nonlinear nature of the equation are used.

Five examples with varying parameters have been taken to elaborate the efficacy of the method.

The numerical results obtained comply with the nature of solution of Burgers–Fisher equation and are better than results available in the literature.

Method is very efficient, less complex and can be extended to higher dimensional partial differential equations.
Notes
Acknowledgements
The authors thank the anonymous referees for their useful and timely suggestions. The corresponding author would like to thank Prof R.C. Mittal for being a great mentor and guiding towards the right path. His support, patience and time is truly acknowledged.
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