Traveling wave solutions of the time-delayed generalized Burgers-type equations

Tang et al. SpringerPlus (2016) 5:2094 DOI 10.1186/s40064-016-3765-1 Open Access RESEARCH Traveling wave solutions of the time‑delayed generalized Burgers‑type equations Bo Tang1,2*, Yingzhe Fan1,3, Xuemin Wang4, Jixiu Wang1 and Shijun Chen1 *Correspondence: 1 School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang 441053, China Full list of author information is available at the end of the article Abstract Background: Recently, nonlinear time-delayed evolution equations have received considerable interest due to their numerous applications in the areas of physics, biology, chemistry and so on. In this paper, we obtain traveling wave solutions by using the extended  Methods: G′ G -expansion method. Results: Based on the method, we get many solutions of the time-delayed generalized Burgers-type equations.  ′ Conclusions: The results reveal that the extended GG -expansion method is direct, effective and can be used for many other nonlinear time-delayed evolution equations.  ′ Keywords: Nonlinear time-delayed evolution equations, Extended GG -expansion method, Traveling wave solution Background In recent years, theory and numerical analysis of nonlinear time-delayed evolution equations have received considerable interest due to their numerous applications in the areas of physics, biology, chemistry and so on. For better studying the nonlinear physical phenomena of nonlinear time-delayed evolution equations, the solution is much involved. In the past, several analytical and numerical methods have been used to find solutions of nonlinear partial differential equations, such as homotopy perturbation method (Kumar and Singh 2009; Kumar et al. 2012; He 1999), Laplace transform (Kumar 2014), variational iteration method (He 1997; He and Wu 2007; Tang et al. 2014), residual power series method (RPSM for short) (Kumar et al. 2016b; Yao et al. 2015), auxiliary equation method (Sirendaoreji 2003; Tang et al.  Yomba 2004), homotopy analysis method  2016; ′ (Yin et al. 2015; Kumar et al. 2016a), GG -expansion method (Wang et al. 2008; Zhang et al. 2010; Tang et al. 2011; Islam et al.2014;  Khan and Akbar 2014) and so on. G′ In this paper, we apply the extended G -expansion method to obtain traveling wave solutions of the following time-delayed generalized Burgers-type equations (Kar et al. 2003): © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Tang et al. SpringerPlus (2016) 5:2094 Page 2 of 16 ••  The time-delayed generalized Burgers equation: τ vtt + vt + pvs vx − vxx = 0. where p, s are constants and τis a time-delayed constant. ••  The time-delayed generalized Burgers-Fisher equation: τ vtt + (1 − τ fv )vt = vxx − pvs vx + f (v), f (v) = qv(1 − vs ).  paper is organized as follows: in “Methods” section, the main steps of extended This G′ G -expansion method for obtaining traveling wave solutions of nonlinear time-delayed evolution equation are given. In “Results” section, we construct traveling solutions of the time-delayed generalized Burgers-type equation. Some conclusions are given in “Conclusions” section. Methods Considering the following nonlinear evolution equation: (1) P(v, vt , vx1 , vx2 , vx3 , . . .) = 0, where P is a polynomial in v = v(x1 , x2 , x3 , . . . , t) and its various partial derivatives. Step 1 By means of the traveling wave transformation η = k1 x1 + k2 x2 + k3 x3 + · · · + ht + η0 , v = V (η), (2) where the coefficients ki, h are constants. Equation (1) can be transformated as follows: ′′ P(V (η), V ′ (η), V (η), . . .) = 0. Step 2 (3) We suppose that the Eq. (3) has the following solution: V (η) = n  l=−n al  G′ G l , (4) where al are constants to be determined later, and G(η) satisfies the following equation: G ′′ (η) + αG ′ (η) + βG(η) = 0, (5) where α and β are arbitrary constants. Based on Eq. (5), we have  �√ � �√ �  α 2 −4βη α 2 −4βη √  C1 sinh +C2 cosh  2 2 2  α −4β   �√ � � , �√  − α2 +  2 α 2 −4βη α 2 −4βη   sinh +C C cosh 2 1  2 2 �√ �√ � �  G ′ (η)  4β−α 2 η 4β−α 2 η √ = −C1 sin +C2 cos 2 2 2 4β−α   G(η)  �√ �√ � � , − α2 +  2  4β−α 2 η 4β−α 2 η  C1 cos +C2 sin  2 2     C2 − α , C1 +C2 η 2 α 2 − 4β > 0, α 2 − 4β < 0. α 2 − 4β = 0. Tang et al. SpringerPlus (2016) 5:2094 Page 3 of 16 Step 3 Determine the degree n in Eq. (3) by use of homogenous balanced principle (Abdel Rady et al. 2010; Fan and Zhang 1998a, b; Senthilvelan 2001; Zhao and Tang 2002; Fan 2000; Eslami et al. 2014), namely balancing the highest order derivatives and nonlinear terms in Eq. (3). Step 4 Substituting Eqs. (4) and (5) into Eq. (3) and clearing the denominator and col ′ lecting all terms with the same order of GG together, then setting each coefficient of  ′ l G to zero, we get a system of under-determined algebraic equations for ki , h and al. G Step 5 Solving the algebraic equations in Step 4 by Maple (www.maplesoft.com), we can finally get traveling wave solutions of Eq. (1). Results  ′ In this section, we apply the extended GG -expansion method to obtain traveling wave solutions of the time-delayed generalized Burgers-type equations. Solutions to the time‑delayed generalized Burgers equation We consider the following time-delayed generalized Burgers equation: τ vtt + vt + pvs vx − vxx = 0. (6) By using transformations v(x, t) = V (η) and η = k(x − ωt), Eq. (6) can be reduced as follows: (τ ω2 − 1)k 2 V ′′ − kωV ′ + pkV s V ′ = 0. (7) Balancing V ′′ with V s V ′ gives n = 1s which is not an integer as s � = 1. So we use a trans1 formation V = W s to change Eq. (7) into the form:     1 − 1 W ′2 − kωW ′ W + pkW ′ W 2 = 0. (τ ω2 − 1)k 2 W ′′ W + (8) s We suppose that the solutions of (8) have the form (4) and (5), so n  G′ G  l−1 G ′′ G − G ′2 G2 l=−n    ′   ′ 2  n  G G ′ l−1 G + β +α , =− al G G G l=−n  ′   ′ 2 2  ′ l−2  n  G G G ′′ + β +α W (η) = al G G G l=−n       ′   ′ 2   n  G′ G G G ′ l−1 β +α α+2 + . + al G G G G ′ W (η) = al  l=−n From above two equations, we can get the degrees of W ′′ W and W ′ W 2 are 2n + 2 and 3n + 1 respectively. Balancing W ′′ W and W ′ W 2 in Eq. (8) yields 2n + 2 = 3n + 1, namely n = 1. Therefore Eq. (8) have the following solutions: Tang et al. SpringerPlus (2016) 5:2094 W (η) = Page 4 of 16 1  l=−1 al  G′ G l (9) . Substituting Eqs. (9) and (5) into Eq. (8), we get a set of under-determined algebraic equations for al (l = 0, ±1), k, ω, α and β.  G′ G 4 : 2(τ ω2 − 1)ka21 +  G′ G 3 : 3(τ ω2 − 1)kαa21 + 2(τ ω2 − 1)ka1 a0 k(τ ω2 − 1)(1 − s)a21 − pa31 = 0, s 2kα(τ ω (...truncated)