# Exponential B-spline collocation method for solving the generalized Newell-Whitehead-Segel equation

Volume 20, Issue 4, pp 313--324
Publication Date: March 05, 2020 Submission Date: October 16, 2019 Revision Date: December 30, 2019 Accteptance Date: January 16, 2020
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### Authors

Imtiaz Wasim - Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan. Muhammad Abbas - Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan. Muhammad Kashif Iqbal - Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan. Afzaal Mubashir Hayat - Department of Mathematics, National College of Business Administration $\&$ Economics, Lahore, 54660, Pakistan.

### Abstract

In this work, we present a collocation method based on exponential basis spline functions for solving generalized Newell-Whitehead-Segel equation. The time derivative is discretized by finite difference scheme and the exponential basis spline functions are employed to interpolate spatial derivatives. The convergence and stability of the proposed algorithm are established. Numerical results demonstrate the accuracy of the proposed method.

### Share and Cite

##### ISRP Style

Imtiaz Wasim, Muhammad Abbas, Muhammad Kashif Iqbal, Afzaal Mubashir Hayat, Exponential B-spline collocation method for solving the generalized Newell-Whitehead-Segel equation, Journal of Mathematics and Computer Science, 20 (2020), no. 4, 313--324

##### AMA Style

Wasim Imtiaz, Abbas Muhammad, Iqbal Muhammad Kashif, Hayat Afzaal Mubashir, Exponential B-spline collocation method for solving the generalized Newell-Whitehead-Segel equation. J Math Comput SCI-JM. (2020); 20(4):313--324

##### Chicago/Turabian Style

Wasim, Imtiaz, Abbas, Muhammad, Iqbal, Muhammad Kashif, Hayat, Afzaal Mubashir. "Exponential B-spline collocation method for solving the generalized Newell-Whitehead-Segel equation." Journal of Mathematics and Computer Science, 20, no. 4 (2020): 313--324

### Keywords

• exponential B-spline collocation method
• convergence
• stability

•  65M70
•  65Z05
•  65N12
•  65D05
•  65D07

### References

• [1] A. Aasaraai, Analytic solution for Newell--Whitehead--Segel equation by differential transform method, Middle-East J. Sci. Res., 10 (2011), 270--273

• [2] M. Abbas, A. A. Majid, A. I. M. Ismail, A. Rashid, The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Appl. Math. Comput., 239 (2014), 74--88

• [3] E. Babolian, J. Saeidian, Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1984--1992

• [4] C. Clavero, J. C. Jorge, F. Lisbona, Uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, J. Comput. Appl. Math., 154 (2003), 415--429

• [5] C. de Boor, On the convergence of odd degree spline interpolation, J. Approximation Theory, 1 (1968), 452--463

• [6] R. Ezzati, K. Shakibi, Using Adomian's decomposition and multiquadric quasi-interpolation methods for solving Newell–Whitehead equation, Procedia Comput. Sci., 3 (2011), 1043--1048

• [7] A. V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics, World Scientific Publishing Co., River Edge (1998)

• [8] A. A. Golovin, A. A. Nepomnyashchy, General Aspects of Pattern Formation, in: Self-assembly, pattern formation and growth phenomena in nano-systems, 2007 (2007), 1--54

• [9] C. A. Hall, On error bounds for spline interpolation, J. Approximation Theory, 1 (1968), 209--218

• [10] M. K. Iqbal, M. Abbas, I. Wasim, New cubic B-spline approximation for solving third order Emden–Flower type equations, Appl. Math. Comput., 331 (2018), 319--333

• [11] H. Kheiri, N. Alipour, R. Dehghani, Homotopy analysis and Homotopy Padé methods for the modified Burgers-Korteweg-de Vries and the Newell-Whitehead equations, Math. Sci. Q. J., 5 (2011), 33--50

• [12] J. E. Macias-Diaz, J. Ruiz-Ramirez, A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell-Whitehead Segel equation, Appl. Numer. Math., 61 (2011), 630--640

• [13] S. S. Nourazar, M. Soori, A. Nazari-Golshan, On the exact solution of Newell--Whitehead-Segel equation using the homotopy perturbation method, Aust. J. Basic Appl. Sci., 5 (2011), 1400--1411

• [14] J. Patade, S. Bhalekar, Approximate analytical solutions of Newell--Whitehead--Segel equation using a new iterative method, World J. Modell. Simul., 11 (2015), 94--103

• [15] W. Rudin, Principles of Mathematical analysis, McGraw-Hill Book Co., New York-Auckland-Düsseldorf (1976)

• [16] UC. San Diego, Rayleigh-Benard Convection, Department of Physics, UC. San Diego (2009)

• [17] R. A. Van Gorder, K. Vajravelu, A variational formulation of the Nagumo reaction diffusion equation and the Nagumo telegraph equation, Nonlinear Anal. Real World Appl., 11 (2010), 2957--2962

• [18] A. Voigt, Asymptotic behavior of solutions to the Allen-Cahn equation in spherically symmetric domains, Caesar-Preprint, 2001 (2001), 1--8

• [19] I. Wasim, M. Abbas, M. Amin, Hybrid b-spline collocation method for solving the generalized Burgers-Fisher and Burgers-Huxley equations, Math. Probl. Eng., 2018 (2018), 18 pages

• [20] I. Wasim, M. Abbas, M. K. Iqbal, Numerical solution of modified forms of Camassa-Holm and Degasperis-Procesl equations via quartic B-spline collocation method, Commun. Math. Appl., 9 (2018), 393--409

• [21] W. K. Zahra, W. A. Ouf, M. S. El-Azab, Cubic $B$-spline collocation algorithm for the numerical solution of Newell Whitehead Segel type equations, Electron. J. Math. Anal. Appl., 2 (2014), 81--100