Research Papers

Complex Wave Solutions to Mathematical Biology Models I: Newell–Whitehead–Segel and Zeldovich Equations

[+] Author and Article Information
Alper Korkmaz

Department of Mathematics,
Çankırı Karatekin University,
Çankırı 18200, Turkey
e-mail: akorkmaz@karatekin.edu.tr

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 29, 2018; final manuscript received May 21, 2018; published online June 18, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(8), 081004 (Jun 18, 2018) (7 pages) Paper No: CND-18-1042; doi: 10.1115/1.4040411 History: Received January 29, 2018; Revised May 21, 2018

Complex and real valued exact solutions to some reaction-diffusion equations are suggested by using homogeneous balance and Sine-Gordon equation expansion method. The predicted solution of finite series of some hyperbolic functions is determined by using some relations between the hyperbolic functions and the trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel–Whitehead–Segel (NWSE) and Zeldovich equations (ZE) are solved explicitly. Some complex valued solutions are depicted in real and imaginary components for some particular choice of parameters.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Gilding, B. H. , and Kersner, R. , 2001, Traveling Waves in Nonlinear Diffusion-Convection-Reaction, University of Twente, Numerical Analysis and Computational Mechanics (NACM) Group, Enschede, The Netherlands. [PubMed] [PubMed]
Newell, A. C. , and Whitehead, J. A. , 1969, “ Finite Bandwidth, Finite Amplitude Convection,” J. Fluid Mech., 38(2), pp. 279–303. [CrossRef]
Segel, A. , 1969, “ Distant Side-Walls Cause Slow Amplitude Modulation of Cellular Convection,” J. Fluid Mech., 38(1), pp. 203–224. [CrossRef]
Meiron, D. , and Newell, A. C. , 1985, “ The Shape of Stationary Dislocations,” Phys. Lett. A, 113(5), pp. 289–292. [CrossRef]
Nepomnyashcy, A. A. , and Pismen, L. M. , 1991, “ Singular Solutions of the Nonlinear Phase Equation in Pattern-Forming Systems,” Phys. Lett. A, 153(8–9), pp. 427–430. [CrossRef]
Malomed, B. A. , Nepomnyashchy, A. A. , and Tribelsky, M. , 1990, “ Domain Boundaries in Convection Patterns,” Phys. Rev. A, 42(12), p. 7244. [CrossRef] [PubMed]
Malomed, B. , 1998, “ Stability and Grain Boundaries in the Dispersive Newell-Whitehead-Segel Equation,” Phys. Scr., 57(1), pp. 115–117. [CrossRef]
Graham, R. , 1996, “ Systematic Derivation of a Rotationally Covariant Extension of the Two-Dimensional Newell-Whitehead-Segel Equation,” Phys. Rev. Lett., 76(12), p. 2185. [CrossRef] [PubMed]
Zemskov, E. P., 2014, “ Turing Patterns and Newell-Whitehead-Segel Amplitude Equation,” Phys.-Usp., 57(10), p. 1035. [CrossRef]
Caraballo, T. , Crauel, H. , Langa, J. A. , and Robinson, J. C. , 2007, “ The Effect of Noise on the Chafee-Infante Equation: A Nonlinear Case Study,” Proc. Am. Math. Soc., 135(2), pp. 373–382. [CrossRef]
Saravanan, A. , and Magesh, N. , 2013, “ A Comparison Between the Reduced Differential Transform Method and the Adomian Decomposition Method for the Newell-Whitehead-Segel Equation,” J. Egyptian Math. Soc., 21(3), pp. 259–265. [CrossRef]
Prakash, A. , and Manoj, K. , 2016, “ He's Variational Iteration Method for the Solution of Nonlinear Newell-Whitehead-Segel Equation,” J. Appl. Anal. Comput., 6(3), pp. 738–748.
Danilov, V. G. , Maslov, V. P. , and Volosov, K. V. , 1995, Mathematical Modelling of Heat and Mass Transfer Processes, Kluwer Academic Publishers, Dordrecht, The Netherlands. [CrossRef]
Korkmaz, A. , 2017, “ Exact Solutions to (3 + 1) Conformable Time Fractional Jimbo-Miwa, Zakharov-Kuznetsov and Modified Zakharov-Kuznetsov Equations,” Commun. Theor. Phys., 67(5), pp. 479–482. [CrossRef]
Kumar, D. , Singh, J. , and Baleanu, D. , 2018, “ A New Numerical Algorithm for Fractional Fitzhugh-Nagumo Equation Arising in Transmission of Nerve Impulses,” Nonlinear Dyn., 91(1), pp. 307–317. [CrossRef]
Singh, J. , Kumar, D. , Qurashi, M. A. , and Baleanu, D. , 2017, “ A Novel Numerical Approach for a Nonlinear Fractional Dynamical Model of Interpersonal and Romantic Relationships,” Entropy, 19(7), p. 375. [CrossRef]
Kumar, D. , Singh, J. , and Baleanu, D. , 2017, “ A New Analysis for Fractional Model of Regularized Long Wave Equation Arising in Ion Acoustic Plasma Waves,” Math. Methods Appl. Sci., 40(15), pp. 5642–5653. [CrossRef]
Kumar, D. , Singh, J. , Baleanu, D. , and Sushila, J. , 2018, “ Analysis of Regularized Long-Wave Equation Associated With a New Fractional Operator With Mittag-Leffler Type Kernel,” Phys. A: Stat. Mech. Appl., 492, pp. 155–167. [CrossRef]
Singh, J. , Rashidi, M. M. , Sushila, J. , and Kumar, D. , 2017, “ A Hybrid Computational Approach for Jeffery–Hamel Flow in Non-Parallel Walls,” Neural Comput. Appl., pp. 1–7.
Korkmaz, A. , 2017, “ Exact Solutions of Space-Time Fractional Ew and Modified Ew Equations,” Chaos, Solitons Fractals, 96, pp. 132–138. [CrossRef]
Korkmaz, A. , 2018, “ On the Wave Solutions of Conformable Fractional Evolution Equations,” Commun. Ser. A1: Math. Stat., 67(1), pp. 68–79.
Rezazadeh, H. , Korkmaz, A. , Eslami, M. , Vahidi, J. , and Asghari, R. , 2018, “ Traveling Wave Solution of Conformable Fractional Generalized Reaction Duffing Model by Generalized Projective Riccati Equation Method,” Opt. Quantum Electron., 50(3), p. 150. [CrossRef]
Yan, C. , 1996, “ A Simple Transformation for Nonlinear Waves,” Phys. Lett. A, 224(1–2), pp. 77–84. [CrossRef]


Grahic Jump Location
Fig. 1

Real and Imaginary parts of the particular solution u1(z,t): (a) real component and (b) imaginary component

Grahic Jump Location
Fig. 2

Real and Imaginary parts of the particular solution u25(z,t): (a) real component and (b) imaginary component




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In