In this study, a computational method referred to as Perturbation

Iteration Transform Method (PITM), which is a combination of the

conventional Laplace Transform Method (LTM) and the Perturbation

Iteration Algorithm (PIA) is applied for the solution of Newell-Whitehead-

Segel Equations (NWSEs). Three unique examples are considered and the

results obtained are compared with their exact solutions graphically. Also,

the results agree with those obtained via other semi-analytical methods viz:

New Iterative Method and Adomian Decomposition Method. This present

method proves to be very efficient and reliable. Mathematica 10 is used for

all the computations in this stud

Akinlabi, G. O.

Edeki, S.O.

Journal of Mathematics and Statistics

arXiv

Covenant University Repository

   © 2017 Grace O. Akinlabi and Sunday O. Edeki. This open access article is distributed under a Creative Commons Attribution (CC-BY) 3.0 license. Journal of Mathematics and Statistics    Original Research Paper Perturbation Iteration Transform Method for the Solution of Newell-Whitehead-Segel Model Equations  Grace O. Akinlabi and Sunday O. Edeki  Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria  Article history Received: 16-09-2016 Revised: 16-12-2016 Accepted: 17-02-2017  Corresponding Author: Grace O. Akinlabi Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria Email: grace.akinlabi@covenantuniversity.edu.ng Abstract: In this study, a computational method referred to as Perturbation Iteration Transform Method (PITM), which is a combination of the conventional Laplace Transform Method (LTM) and the Perturbation Iteration Algorithm (PIA) is applied for the solution of Newell-Whitehead-Segel Equations (NWSEs). Three unique examples are considered and the results obtained are compared with their exact solutions graphically. Also, the results agree with those obtained via other semi-analytical methods viz: New Iterative Method and Adomian Decomposition Method. This present method proves to be very efficient and reliable. Mathematica 10 is used for all the computations in this study.  Keywords: Newell-Whitehead-Segel Equation, Perturbation Iteration Technique, Laplace Transform, PITM  Introduction The numerical and analytical approximations of Partial Differential Equation (PDE) problems have always been an active field of study in Physics, Mathematics and Engineering. Many researchers have proposed several approaches to solve different PDE problems. For instance, He (2005) solved some wave equations with the Homotopy Perturbation Method (HPM). Akinlabi and Edeki (2016) also solved initial-value wave-like models using the modified Differential Transform Method (DTM). Likewise, Edeki et al. (2016) considered the numerical and the analytic solutions of time-fractional linear Schrödinger equations. In this study, we are concerned with the solution of the Newell-Whitehead-Segel Equations (NWSEs) expressed as follows:  ( ) ( ) ( ) ( )nt xxx,t r x,t s x,t t x,tµ µ µ µ= + −   (1)  Subject to:  ( ) ( )x,0 xµ ϕ=   (2)  where, r,s,t∈ℝ  with r>0 and n+∈ℤ . Equation (1) was derived by Newell and Whitehead (1969; Segel, 1969) and it has been used in modeling various forms of problems that arise from fluid mechanics. It has applications in Chemical, Bio-Engineering and Mechanical Engineering, etc. This equation has been applied to a number of problems. An example of such is in the description of traveling waves by Malomed (1996). Several authors have proposed different methods of solving this equation in the past years. Saravanan and Magesh (2013) solved two nonlinear NWS equations with both the reduced DTM and the Adomian Decomposition Method (ADM). A comparative test was carried out between these two methods where it was shown that the reduced DTM requires less computational work than the ADM. Prakash and Kumar (2016) used the He's Variational Iteration Method to solve non-linear NWSEs, Macías-Díaz and Ruiz-Ramírez (2011) considered the generalised NWSEs using a non-standard symmetry-preserving method. Others are Aasaraai (2011; Ezzati and Shakibi, 2011; Nourazar et al., 2011) and so on. The idea of using Perturbation Iteration Transform Method (PITM) to solve PDE problems was first pioneered by (Khalid et al., 2016), where the method was used to solve Klein-Gordon equations. The basic idea of this approach is that the PIA and the Laplace Transform (LT) Method are combined to approximate models arising from different fields. The remaining part of the paper is structured as follows: In section 2 and 3, we review the PIA and the PITM respectively. Section 4 is on the application of the PITM to three cases of NWS equations to show its efficiency. The concluding remark is given in section 5. Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ Perturbation Iteration Algorithm (Aksoy and Pakdemirli, 2010) In this section, we illustrate how the PIA works. If we develop a perturbation algorithm by considering one correction term in the perturbed expansion and the correction terms of the first derivatives in the Taylor series expansion, say n = 1 and m = 1. The algorithm will be referred to as PIA(1,1). Now, let us consider a second order differential equation:  ( ), , , 0F µ µ µ ε′′ =ɺ   (3)  where, ( ),x tµ µ= , tµµ∂=∂ɺ , 22xµµ∂′′ =∂ and ε is the newly introduced perturbed parameter. And if we consider only one correction term in the expansion:  ( )1n n c nµ µ ε µ+ = +   (4)   Thus, substituting (4) in (3) and expanding such in Taylor series with first derivatives will give:  ( ) ( ) ( )( ) ( )( ) ( ) ( ), , ,0 , , ,0, , ,0 , , ,0 , , ,0 0c nc nc nF FFF Fµµµ εµ µ µ µ µ µ ε µµ µ µ ε µµ µ µ ε µ µ µ µ ε′′′′ ′′ + ′′ ′′+′′ ′′+ + =ɺɺ ɺ ɺɺɺ ɺ  (5)  where, µ = µ(x,t), FFµ µ∂=∂ɺ ɺ, FFµ µ′′∂=′′∂, FFµ µ∂=∂, FFε ε∂=∂ and ε the perturbation parameter to be evaluated at  ε = 0. Reorganizing (5), we have:  ( ) ( ) ( )c c cn n nFFF FF F Fεµ µµ µ µεµ µ µ′′+′′+ = − −ɺ ɺ ɺɺ   (6)  With a guessed value, u0, the term (µc)0 is obtained from (6) and then put in (4) to evaluate µ1. We continue the iterative process using (6) and (4) till the result(s) are satisfied.  Perturbation Iteration Transform Method (Khalid et al., 2016) To demonstrate the main idea of this method, we consider a PDE with boundary conditions of the form:  ( ) ( ) ( ) ( ) ( ), , , , ,A x t B x t C x t D x t h x tµ µ µ µ+ + + =   (7)  with the initial condition:  ( ) ( ),0x xµ ϕ=   (8) where, At∂=∂ is the first order linear differential operator, 22Bx∂=∂is the second order linear differential operator, C,D are the linear and nonlinear terms and h(x,t) is the source term. Taking the LT of both sides of (7) gives:  ( ) ( ) ( )( ) ( ), , ,, ,L A x t +L B x t +L C x t+L D x t =L h x tµ µ µµ                  (9)  Which on using the differential property of LT yield:  ( ) ( ) ( )( ) ( ) ( )1, ,1, , ,xL A x t = L h x ts sL B x t C x t D x tsϕµµ µ µ  +     −  + +    (10)  Applying the Inverse LT to both sides of (10) gives:  ( ) ( )( ) ( ) ( )1, ,1, , ,x t =H x tL L B x t C x t D x tsµµ µ µ−  −  + +      (11)  where, H(x,t) is the term gotten from the imposed initial condition with the source term. So, by using the PITM (11) becomes:  ( ) ( ) ( )( ) ( ) ( )1, , ,1, , ,cx t H x t x tL L B x t C x t D x t =0sµ µ εµ µ µ ε−− + +  + +      (12)  Thus:  ( ) ( ) ( )( ) ( ) ( )1, ,,1, , ,cH x t x tx t =L L B x t C x t D x tsµµεµ µ µ−− −  + +      (13)  This is the combined form of the LTM and the PIA. The initial point, (µc)0 is then obtained from (13) and then substituted in (4) to obtain µ1. The process is iteratively repeated for µ2, µ3 and so on. The approximate solution is thus obtained by:  ( ) ( ) ( ) ( ) ( )0 1 2 3, , , , ,x t x t x t x t x tµ µ µ µ µ= + + + +⋯   (14)  Illustrative and Numerical Examples Here, the proposed method is applied to the following problems. Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ Cases I, II and III Case I: Consider the Newell-Whitehead-Segel equation (Patade and Bhalekar, 2015):  ( ) ( ) ( ) ( )2, 5 , 2 , ,t xxx t x t x t x tµ µ µ µ= + +   (15)  Subject to:   ( ),0xµ λ=      (16)  With the exact solution:  ( )2t2t2e2 1 eλλ+ −     (17)  Solution Procedure to Case I Taking the LT of both sides of (15), we get:  ( ) ( ) ( ) ( )21, 5 , 2 , ,xxL x t = + L x t x t x ts sλµ µ µ µ   + +      (18)  Applying the ILT to both sides of (18) gives:  ( ) ( ) ( ) ( )1 21, 5 , 2 , ,xxx t = L L x t x t x tsµ λ µ µ µ−   + + +     (19)  Now, by the PITM (19) becomes:  ( ) ( )( ) ( ) ( )1 2, ,15 , 2 , ,cxxx t x tL L x t x t x t =0sµ λ µ εµ µ µ ε−− +  − + +     (20)  Thus:  ( ) ( )( ) ( ) ( )1 2,,15 , 2 , ,cxxx tx t =L L x t x t x tsµ λµεµ µ µ−− +  + + +     (21)  This implies that:  ( )( )( )( )0213 2 3 3 3 42 2 223 3 2 4 2 5 2 4 2 5 336 3 4 4 5 4 6 4 7 4 6 57 5 6 6 7 6 7 7 7 8,, 24 4, 23 3 34 2 2 4 2 4,3 3 3 5 3 58 4 16 28 6 5 3 63 332 8 863 9 21 63 63x tx t t tt t tx t t tt t t t t tx tt t t t t tt t t t tµ λµ λ λλ λ λµ λ λλ λ λ λ λ λµλ λ λ λ λ λλ λ λ λ λ== += + + + += + + + + ++ + + + + ++ + + + +⋮ Therefore, the solution µ(x,t) is given by:  ( ) ( ) ( ) ( ) ( )0 1 2 33 2 3 32 2 2 23 4 3 3 2 4 2 5 2 4 25 3 6 3 4 4 5 4 6 4 7 46 5 7 5 6 6 7 6 7 7 7 8, , , , ,4 42 23 34 2 2 4 23 3 3 3 5 34 8 4 165 8 6 5 3 632 32 8 83 63 9 21 63 63x t x t x t x t x tt tt t t tt t t t t tt t t t t tt t t t t tµ µ µ µ µλ λλ λ λ λ λλ λ λ λ λ λλ λ λ λ λ λλ λ λ λ λ λ= + + + += + + + + + ++ + + + + ++ + + + + ++ + + + + + +⋯⋯  (22)  Case II: Consider the Newell-Whitehead-Segel equation (Patade and Bhalekar, 2015):  ( ) ( ) ( ) ( )2, , 2 , 3 ,t xxx t x t x t x tµ µ µ µ= + −   (23)  Subject to:  ( ),0xµ λ=     (24)  With the exact solution:  ( )2t2t2e2 3 1 eλλ− + −    (25)  Solution Procedure to Case II Taking the LT of both sides of (23), we get:  ( ) ( ) ( ) ( )21, , 2 , 3 ,xxL x t = + L x t x t x ts sλµ µ µ µ   + −        (26)  Applying the ILT to both sides of (26) gives:  ( ) ( ) ( ) ( )1 21, , 2 , 3 ,xxx t = +L L x t x t x tsµ λ µ µ µ−   + −       (27)  Now, by the PITM (27) becomes:  ( ) ( )( ) ( ) ( )1 2, ,1, 2 , 3 ,cxxx t x tL L x t x t x t =0sµ λ µ εµ µ µ ε−− +  − + −       (28)  Thus:  ( ) ( )( ) ( ) ( )1 2,,1, 2 , 3 ,cxxx tx t =L L x t x t x tsµ λµεµ µ µ−− +  + + −       (29)  This implies that: Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ ( )( )( )( )0212 2 2 3 2 3 3 3 423 5 2 5 33 2 4 2 4 334 4 5 4 7 46 3 6 4 6 57 5 7 6 7 7 7 86 6,, 2 3, 2 3 4 12 94 12 36, 2 2 63 5 59 27 488 36 542 5 7288 648 648 243277 7 7 7x tx t t tx t t t t t tt t tx t t t tt t tt t tt t t ttµ λµ λ λµ λ λ λ λ λλ λ λµ λ λ λλ λ λλ λ λλ λ λ λλ== −= − − + −= − − − + ++ − − − − ++ − − + −⋮  Therefore, the solution µ(x,t) is given by:  ( ) ( ) ( ) ( ) ( )0 1 2 32 2 2 2 3 2 3 3 3 43 5 2 5 33 2 4 2 4 34 4 5 4 7 46 3 6 4 6 57 5 7 6 7 7 7 86 6, , , , ,2 3 2 3 4 12 94 12 362 2 63 5 59 27 488 36 542 5 7288 648 648 243277 7 7 7x t x t x t x t x tt t t t t t tt t tt t tt t tt t tt t t ttµ µ µ µ µλ λ λ λ λ λ λ λλ λ λλ λ λλ λ λλ λ λλ λ λ λλ= + + + += + − + − − + −+ − − − + ++ − − − − ++ − − + − +⋯⋯    (30)  Case III: Consider the Newell-Whitehead equation (Patade and Bhalekar, 2015):  ( ) ( ) ( ) ( )3, , 2 , 3 ,t xxx t x t x t x tµ µ µ µ= + −     (31)  Subject to:  ( )222,03xx xexe eµ =+    (32)  Solution Procedure to Case III Taking the LT of both sides of (31), we get:  ( )( ) ( ) ( )2231 2,31, 2 , 3 ,xx xxxeL x t =s e e+ L x t x t x tsµµ µ µ   + + −     (33)  Applying the ILT to both sides of (33) gives:  ( )( ) ( ) ( )221 32,31, 2 , 3 ,xx xxxex t =e e+L L x t x t x tsµµ µ µ−+  + −       (34)  Now, by the PITM (34) becomes:   ( ) ( )( ) ( ) ( )221 32, ,31, 2 , 3 ,xcx xxxex t x te eL L x t x t x t =0sµ µ εµ µ µ ε−− ++  − + −       (35) Thus:  ( )( )( ) ( ) ( )221 32,3,1, 2 , 3 ,xx xcxxex te ex t =L L x t x t x tsµµεµ µ µ−− ++  + + −       (36)  This implies that:  ( ) ( )( )( )( )( )( ) ( )( )( )( ) ( )( )( )( )62230 1 32 22 222322 222 223 22 22 23 3 432 22 4 62 3 4 3323 622, ,34 22 62 22 433 1 9,1 13 6 22 6 3,1xxx x x xx x xxx x x xx x x xxx x x xx x xx xx x x x xx eex t x t te e e e e e eet te e e ee e e ee te e e e e e t  e tx te e e e e e e tx teµ µµµ= = −+ +++ −+ + + + + − + + += −+ +− + − +=+−( )( )( )( )( )( )( )( )( )33 2 5 3 2 73 32 28 1225 2 9 7 2 113 32 214 169 1332189 11 20 11 243 15 1 14 1243 1 6561 12 1 22 1656126 1x x x x xx xx x x xx xxx e e e t  e e te e e e t  e e te e e te− + +−+ ++ ++ −+ +++⋮  Therefore, the solution µ(x,t) is given by:  ( ) ( ) ( ) ( ) ( )( )( )( )( )( ) ( )( )( )( )0 1 2 32 26 222 2333 22 22 222 2 232223 2 42 22 3 433 43226, , , , ,4 222 2 632 2 3 12 43 13 6 22 6 391x x xxx xx x x xx x x xx x x xx xxx x x x xx x x xxxx t x t x t x t x t e e e ee et t te e e ee e e ee e  e e te ee te e e e e e e e e e e teµ µ µ µ µ= + + + ++= − + −+ ++ + + ++ + − + + + + − + − +− ++⋯( )( )( )( )( )( )( )( )( )( )( )3633 2 5 3 2 73 32 28 1225 2 9 7 2 113 32 214 169 13321819 11 20 11 243 15 1 14 1243 1 6561 12 1 22 1656126 1xxx x x x xx xx x x xx xxxte e e e t  e e te e e e t  e e te e e te+− + +− −+ ++ ++ −+ ++ ++⋯   (37) Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■   Fig. 1. Solution graph for case I    Fig. 2. Solution graph for case II     Fig. 3. Solution graph for case III  Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ Figure 1 is for the exact solution and the PITM (approximate solution) of case I. Figure 2 is for exact solution and the PITM (approximate solution) of case II. Figure 3 is for exact solution and the PITM (approximate solution) of case III. Discussion of Results In this subsection, we present the graphs for the exact solution and the PITM (approximate solution) for the three cases. Conclusion In this study, the solutions (roots) of the Newell-Whitehead-Segel models are gotten using Perturbation Iteration Transform Method as a proposed computational method. The results are obtained with less computational time and are compared graphically with their exact solutions. In addition, these results are in good agreement with those by Patade and Bhalekar (2015) using other semi-analytical method: Adomian Decomposition Method and New Iterative Method. We therefore, propose this method for the solutions of both linear and non-linear PDEs in other aspects of applied sciences. Acknowledgement The authors wish to sincerely thank Covenant University (CUCRID) for the provision of enabling environment and finance. In addition, the authors thank the anonymous referees for their valuable contributions. Authors’ Contributions The concerned authors: GOA and SOE contributed significantly to this work. Both authors approved the final manuscript for publication. Ethics The authors declare that there exists no conflict of interest regarding this paper. References Aasaraai, A., 2011. Analytic solution for Newell-whitehead-segel equation by differential transform method. Middle East J. Scientific Res., 10: 270-273. Akinlabi, G.O. and S.O. Edeki, 2016. On approximate and closed-form solution method for initial-value wave-like models. Int. J. Pure Applied Math., 107: 449-456. DOI: 10.12732/ijpam.v107i2.14 Aksoy, Y. and M. Pakdemirli, 2010. New perturbation-iteration solutions for Bratu-type equations. Comput. Math. Applic., 59: 2802-2808.  DOI: 10.1016/j.camwa.2010.01.050 Edeki, S.O., G.O. Akinlabi and S.A. Adeosun, 2016. Analytic and numerical solutions of time-fractional linear schrödinger equation. Commun. Math. Applic., 7: 1-10.  Ezzati, R. and K. Shakibi, 2011. Using adomian’s decomposition and multiquadric quasi-interpolation methods for solving Newell-Whitehead equation. Proc. Comput. Sci., 3: 1043-1048.  DOI: 10.1016/j.procs.2010.12.171 He, J.H., 2005. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals, 26: 695-700.  DOI: 10.1016/j.chaos.2005.03.006 Khalid, M., M. Sultana, F. Zaidi and A. Uroosa, 2016. Solving linear and nonlinear Klein-Gordon equations by new perturbation iteration transform method. TWMS J. Applied Eng. Math., 6: 115-125.  Macías-Díaz, J. and J. Ruiz-Ramírez, 2011. A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell-Whitehead-Segel equation. Applied Numerical Math., 61: 630-640.  DOI: 10.1016/j.apnum.2010.12.008  Malomed, B.A., 1996. The Newell-Whitehead-Segel equation for travelling waves. Newell, A. and J. Whitehead, 1969. Finite bandwidth, finite amplitude convection. J. Fluid Mechan., 38: 279-303. DOI: 10.1017/S0022112069000176 Nourazar, S.S., M. Soori and A. Nazari-Golshan, 2011. On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method. Australian J. Basic Applied Sci. Fractals, 5: 1400-1411.  Patade, J. and S. Bhalekar, 2015. Approximate analytical solutions of Newell-Whitehead-Segel equation using a new iterative method. World J. Modell. Simulat., 11: 94-103.  Prakash, A. and M. Kumar, 2016. He’s variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation. J. Applied Anal. Comput., 6: 738-748. DOI: 10.11948/2016048 Saravanan, A. and N. Magesh, 2013. A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation. J. Egypt. Math. Society, 21: 259-265. DOI: 10.1016/j.joems.2013.03.004 Segel, L.A., 1969. Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mechan., 38: 203-224.  DOI: 10.1017/S0022112069000127 

Perturbation Iteration Transform Method for the Solution of

Newell-Whitehead-Segel Model Equations

In this study, a computational method referred to as Perturbation Iteration Transform Method (PITM), which is a combination of the conventional Laplace Transform Method (LTM) and the Perturbation Iteration Algorithm (PIA) is applied for the solution of Newell-Whitehead- Segel Equations (NWSEs). Three unique examples are considered and the results obtained are compared with their exact solutions graphically. Also, the results agree with those obtained via other semi-analytical methods viz: New Iterative Method and Adomian Decomposition Method. This present method proves to be very efficient and reliable. Mathematica 10 is used for all the computations in this study

Grace O. Akinlabi

Sunday O. Edeki

ZENODO

   © 2017 Grace O. Akinlabi and Sunday O. Edeki. This open access article is distributed under a Creative Commons Attribution (CC-BY) 3.0 license. Journal of Mathematics and Statistics    Original Research Paper Perturbation Iteration Transform Method for the Solution of Newell-Whitehead-Segel Model Equations  Grace O. Akinlabi and Sunday O. Edeki  Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria  Article history Received: 16-09-2016 Revised: 16-12-2016 Accepted: 17-02-2017  Corresponding Author: Grace O. Akinlabi Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria Email: grace.akinlabi@covenantuniversity.edu.ng Abstract: In this study, a computational method referred to as Perturbation Iteration Transform Method (PITM), which is a combination of the conventional Laplace Transform Method (LTM) and the Perturbation Iteration Algorithm (PIA) is applied for the solution of Newell-Whitehead-Segel Equations (NWSEs). Three unique examples are considered and the results obtained are compared with their exact solutions graphically. Also, the results agree with those obtained via other semi-analytical methods viz: New Iterative Method and Adomian Decomposition Method. This present method proves to be very efficient and reliable. Mathematica 10 is used for all the computations in this study.  Keywords: Newell-Whitehead-Segel Equation, Perturbation Iteration Technique, Laplace Transform, PITM  Introduction The numerical and analytical approximations of Partial Differential Equation (PDE) problems have always been an active field of study in Physics, Mathematics and Engineering. Many researchers have proposed several approaches to solve different PDE problems. For instance, He (2005) solved some wave equations with the Homotopy Perturbation Method (HPM). Akinlabi and Edeki (2016) also solved initial-value wave-like models using the modified Differential Transform Method (DTM). Likewise, Edeki et al. (2016) considered the numerical and the analytic solutions of time-fractional linear Schrödinger equations. In this study, we are concerned with the solution of the Newell-Whitehead-Segel Equations (NWSEs) expressed as follows:  ( ) ( ) ( ) ( )nt xxx,t r x,t s x,t t x,tµ µ µ µ= + −   (1)  Subject to:  ( ) ( )x,0 xµ ϕ=   (2)  where, r,s,t ∈ℝ  with r>0 and n+∈ℤ . Equation (1) was derived by Newell and Whitehead (1969; Segel, 1969) and it has been used in modeling various forms of problems that arise from fluid mechanics. It has applications in Chemical, Bio-Engineering and Mechanical Engineering, etc. This equation has been applied to a number of problems. An example of such is in the description of traveling waves by Malomed (1996). Several authors have proposed different methods of solving this equation in the past years. Saravanan and Magesh (2013) solved two nonlinear NWS equations with both the reduced DTM and the Adomian Decomposition Method (ADM). A comparative test was carried out between these two methods where it was shown that the reduced DTM requires less computational work than the ADM. Prakash and Kumar (2016) used the He's Variational Iteration Method to solve non-linear NWSEs, Macías-Díaz and Ruiz-Ramírez (2011) considered the generalised NWSEs using a non-standard symmetry-preserving method. Others are Aasaraai (2011; Ezzati and Shakibi, 2011; Nourazar et al., 2011) and so on. The idea of using Perturbation Iteration Transform Method (PITM) to solve PDE problems was first pioneered by (Khalid et al., 2016), where the method was used to solve Klein-Gordon equations. The basic idea of this approach is that the PIA and the Laplace Transform (LT) Method are combined to approximate models arising from different fields. The remaining part of the paper is structured as follows: In section 2 and 3, we review the PIA and the PITM respectively. Section 4 is on the application of the PITM to three cases of NWS equations to show its efficiency. The concluding remark is given in section 5. Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ Perturbation Iteration Algorithm (Aksoy and Pakdemirli, 2010) In this section, we illustrate how the PIA works. If we develop a perturbation algorithm by considering one correction term in the perturbed expansion and the correction terms of the first derivatives in the Taylor series expansion, say n = 1 and m = 1. The algorithm will be referred to as PIA(1,1). Now, let us consider a second order differential equation:  ( ), , , 0F µ µ µ ε′′ =ɺ   (3)  where, ( ),x tµ µ= , tµµ∂=∂ɺ , 22xµµ∂′′ =∂ and ε is the newly introduced perturbed parameter. And if we consider only one correction term in the expansion:  ( )1n n c nµ µ ε µ+ = +   (4)   Thus, substituting (4) in (3) and expanding such in Taylor series with first derivatives will give:  ( ) ( ) ( )( ) ( )( ) ( ) ( ), , ,0 , , ,0, , ,0 , , ,0 , , ,0 0c nc nc nF FFF Fµµµ εµ µ µ µ µ µ ε µµ µ µ ε µµ µ µ ε µ µ µ µ ε′′′′ ′′ + ′′ ′′+′′ ′′+ + =ɺɺ ɺ ɺɺɺ ɺ  (5)  where, µ = µ(x,t), FFµ µ∂=∂ɺ ɺ, FFµ µ′′∂=′′∂, FFµ µ∂=∂, FFε ε∂=∂ and ε the perturbation parameter to be evaluated at  ε = 0. Reorganizing (5), we have:  ( ) ( ) ( )c c cn n nFFF FF F Fεµ µµ µ µεµ µ µ′′+′′+ = − −ɺ ɺ ɺɺ   (6)  With a guessed value, u0, the term (µc)0 is obtained from (6) and then put in (4) to evaluate µ1. We continue the iterative process using (6) and (4) till the result(s) are satisfied.  Perturbation Iteration Transform Method (Khalid et al., 2016) To demonstrate the main idea of this method, we consider a PDE with boundary conditions of the form:  ( ) ( ) ( ) ( ) ( ), , , , ,A x t B x t C x t D x t h x tµ µ µ µ+ + + =   (7)  with the initial condition:  ( ) ( ),0x xµ ϕ=   (8) where, At∂=∂ is the first order linear differential operator, 22Bx∂=∂is the second order linear differential operator, C,D are the linear and nonlinear terms and h(x,t) is the source term. Taking the LT of both sides of (7) gives:  ( ) ( ) ( )( ) ( ), , ,, ,L A x t +L B x t +L C x t+L D x t =L h x tµ µ µµ                  (9)  Which on using the differential property of LT yield:  ( ) ( ) ( )( ) ( ) ( )1, ,1, , ,xL A x t = L h x ts sL B x t C x t D x tsϕµµ µ µ  +     −  + +    (10)  Applying the Inverse LT to both sides of (10) gives:  ( ) ( )( ) ( ) ( )1, ,1, , ,x t =H x tL L B x t C x t D x tsµµ µ µ−  −  + +      (11)  where, H(x,t) is the term gotten from the imposed initial condition with the source term. So, by using the PITM (11) becomes:  ( ) ( ) ( )( ) ( ) ( )1, , ,1, , ,cx t H x t x tL L B x t C x t D x t =0sµ µ εµ µ µ ε−− + +  + +      (12)  Thus:  ( ) ( ) ( )( ) ( ) ( )1, ,,1, , ,cH x t x tx t =L L B x t C x t D x tsµµεµ µ µ−− −  + +      (13)  This is the combined form of the LTM and the PIA. The initial point, (µc)0 is then obtained from (13) and then substituted in (4) to obtain µ1. The process is iteratively repeated for µ2, µ3 and so on. The approximate solution is thus obtained by:  ( ) ( ) ( ) ( ) ( )0 1 2 3, , , , ,x t x t x t x t x tµ µ µ µ µ= + + + +⋯   (14)  Illustrative and Numerical Examples Here, the proposed method is applied to the following problems. Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ Cases I, II and III Case I: Consider the Newell-Whitehead-Segel equation (Patade and Bhalekar, 2015):  ( ) ( ) ( ) ( )2, 5 , 2 , ,t xxx t x t x t x tµ µ µ µ= + +   (15)  Subject to:   ( ),0xµ λ=      (16)  With the exact solution:  ( )2t2t2e2 1 eλλ+ −     (17)  Solution Procedure to Case I Taking the LT of both sides of (15), we get:  ( ) ( ) ( ) ( )21, 5 , 2 , ,xxL x t = + L x t x t x ts sλµ µ µ µ   + +      (18)  Applying the ILT to both sides of (18) gives:  ( ) ( ) ( ) ( )1 21, 5 , 2 , ,xxx t = L L x t x t x tsµ λ µ µ µ−   + + +     (19)  Now, by the PITM (19) becomes:  ( ) ( )( ) ( ) ( )1 2, ,15 , 2 , ,cxxx t x tL L x t x t x t =0sµ λ µ εµ µ µ ε−− +  − + +     (20)  Thus:  ( ) ( )( ) ( ) ( )1 2,,15 , 2 , ,cxxx tx t =L L x t x t x tsµ λµεµ µ µ−− +  + + +     (21)  This implies that:  ( )( )( )( )0213 2 3 3 3 42 2 223 3 2 4 2 5 2 4 2 5 336 3 4 4 5 4 6 4 7 4 6 57 5 6 6 7 6 7 7 7 8,, 24 4, 23 3 34 2 2 4 2 4,3 3 3 5 3 58 4 16 28 6 5 3 63 332 8 863 9 21 63 63x tx t t tt t tx t t tt t t t t tx tt t t t t tt t t t tµ λµ λ λλ λ λµ λ λλ λ λ λ λ λµλ λ λ λ λ λλ λ λ λ λ== += + + + += + + + + ++ + + + + ++ + + + +⋮ Therefore, the solution µ(x,t) is given by:  ( ) ( ) ( ) ( ) ( )0 1 2 33 2 3 32 2 2 23 4 3 3 2 4 2 5 2 4 25 3 6 3 4 4 5 4 6 4 7 46 5 7 5 6 6 7 6 7 7 7 8, , , , ,4 42 23 34 2 2 4 23 3 3 3 5 34 8 4 165 8 6 5 3 632 32 8 83 63 9 21 63 63x t x t x t x t x tt tt t t tt t t t t tt t t t t tt t t t t tµ µ µ µ µλ λλ λ λ λ λλ λ λ λ λ λλ λ λ λ λ λλ λ λ λ λ λ= + + + += + + + + + ++ + + + + ++ + + + + ++ + + + + + +⋯⋯  (22)  Case II: Consider the Newell-Whitehead-Segel equation (Patade and Bhalekar, 2015):  ( ) ( ) ( ) ( )2, , 2 , 3 ,t xxx t x t x t x tµ µ µ µ= + −   (23)  Subject to:  ( ),0xµ λ=     (24)  With the exact solution:  ( )2t2t2e2 3 1 eλλ− + −    (25)  Solution Procedure to Case II Taking the LT of both sides of (23), we get:  ( ) ( ) ( ) ( )21, , 2 , 3 ,xxL x t = + L x t x t x ts sλµ µ µ µ   + −        (26)  Applying the ILT to both sides of (26) gives:  ( ) ( ) ( ) ( )1 21, , 2 , 3 ,xxx t = +L L x t x t x tsµ λ µ µ µ−   + −       (27)  Now, by the PITM (27) becomes:  ( ) ( )( ) ( ) ( )1 2, ,1, 2 , 3 ,cxxx t x tL L x t x t x t =0sµ λ µ εµ µ µ ε−− +  − + −       (28)  Thus:  ( ) ( )( ) ( ) ( )1 2,,1, 2 , 3 ,cxxx tx t =L L x t x t x tsµ λµεµ µ µ−− +  + + −       (29)  This implies that: Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ ( )( )( )( )0212 2 2 3 2 3 3 3 423 5 2 5 33 2 4 2 4 334 4 5 4 7 46 3 6 4 6 57 5 7 6 7 7 7 86 6,, 2 3, 2 3 4 12 94 12 36, 2 2 63 5 59 27 488 36 542 5 7288 648 648 243277 7 7 7x tx t t tx t t t t t tt t tx t t t tt t tt t tt t t ttµ λµ λ λµ λ λ λ λ λλ λ λµ λ λ λλ λ λλ λ λλ λ λ λλ== −= − − + −= − − − + ++ − − − − ++ − − + −⋮  Therefore, the solution µ(x,t) is given by:  ( ) ( ) ( ) ( ) ( )0 1 2 32 2 2 2 3 2 3 3 3 43 5 2 5 33 2 4 2 4 34 4 5 4 7 46 3 6 4 6 57 5 7 6 7 7 7 86 6, , , , ,2 3 2 3 4 12 94 12 362 2 63 5 59 27 488 36 542 5 7288 648 648 243277 7 7 7x t x t x t x t x tt t t t t t tt t tt t tt t tt t tt t t ttµ µ µ µ µλ λ λ λ λ λ λ λλ λ λλ λ λλ λ λλ λ λλ λ λ λλ= + + + += + − + − − + −+ − − − + ++ − − − − ++ − − + − +⋯⋯    (30)  Case III: Consider the Newell-Whitehead equation (Patade and Bhalekar, 2015):  ( ) ( ) ( ) ( )3, , 2 , 3 ,t xxx t x t x t x tµ µ µ µ= + −     (31)  Subject to:  ( )222,03xx xexe eµ =+    (32)  Solution Procedure to Case III Taking the LT of both sides of (31), we get:  ( )( ) ( ) ( )2231 2,31, 2 , 3 ,xx xxxeL x t =s e e+ L x t x t x tsµµ µ µ   + + −     (33)  Applying the ILT to both sides of (33) gives:  ( )( ) ( ) ( )221 32,31, 2 , 3 ,xx xxxex t =e e+L L x t x t x tsµµ µ µ−+  + −       (34)  Now, by the PITM (34) becomes:   ( ) ( )( ) ( ) ( )221 32, ,31, 2 , 3 ,xcx xxxex t x te eL L x t x t x t =0sµ µ εµ µ µ ε−− ++  − + −       (35) Thus:  ( )( )( ) ( ) ( )221 32,3,1, 2 , 3 ,xx xcxxex te ex t =L L x t x t x tsµµεµ µ µ−− ++  + + −       (36)  This implies that:  ( ) ( )( )( )( )( )( ) ( )( )( )( ) ( )( )( )( )62230 1 32 22 222322 222 223 22 22 23 3 432 22 4 62 3 4 3323 622, ,34 22 62 22 433 1 9,1 13 6 22 6 3,1xxx x x xx x xxx x x xx x x xxx x x xx x xx xx x x x xx eex t x t te e e e e e eet te e e ee e e ee te e e e e e t  e tx te e e e e e e tx teµ µµµ= = −+ +++ −+ + + + + − + + += −+ +− + − +=+−( )( )( )( )( )( )( )( )( )33 2 5 3 2 73 32 28 1225 2 9 7 2 113 32 214 169 1332189 11 20 11 243 15 1 14 1243 1 6561 12 1 22 1656126 1x x x x xx xx x x xx xxx e e e t  e e te e e e t  e e te e e te− + +−+ ++ ++ −+ +++⋮  Therefore, the solution µ(x,t) is given by:  ( ) ( ) ( ) ( ) ( )( )( )( )( )( ) ( )( )( )( )0 1 2 32 26 222 2333 22 22 222 2 232223 2 42 22 3 433 43226, , , , ,4 222 2 632 2 3 12 43 13 6 22 6 391x x xxx xx x x xx x x xx x x xx xxx x x x xx x x xxxx t x t x t x t x t e e e ee et t te e e ee e e ee e  e e te ee te e e e e e e e e e e teµ µ µ µ µ= + + + ++= − + −+ ++ + + ++ + − + + + + − + − +− ++⋯( )( )( )( )( )( )( )( )( )( )( )3633 2 5 3 2 73 32 28 1225 2 9 7 2 113 32 214 169 13321819 11 20 11 243 15 1 14 1243 1 6561 12 1 22 1656126 1xxx x x x xx xx x x xx xxxte e e e t  e e te e e e t  e e te e e te+− + +− −+ ++ ++ −+ ++ ++⋯   (37) Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■   Fig. 1. Solution graph for case I    Fig. 2. Solution graph for case II     Fig. 3. Solution graph for case III  Grace O. Akinlabi and Sunday O. Edeki / Journal of Mathematics and Statistics 2017, ■■ (■): ■■■.■■■ DOI: 10.3844/jmssp.2017.■■■.■■■  ■■■ Figure 1 is for the exact solution and the PITM (approximate solution) of case I. Figure 2 is for exact solution and the PITM (approximate solution) of case II. Figure 3 is for exact solution and the PITM (approximate solution) of case III. Discussion of Results In this subsection, we present the graphs for the exact solution and the PITM (approximate solution) for the three cases. Conclusion In this study, the solutions (roots) of the Newell-Whitehead-Segel models are gotten using Perturbation Iteration Transform Method as a proposed computational method. The results are obtained with less computational time and are compared graphically with their exact solutions. In addition, these results are in good agreement with those by Patade and Bhalekar (2015) using other semi-analytical method: Adomian Decomposition Method and New Iterative Method. We therefore, propose this method for the solutions of both linear and non-linear PDEs in other aspects of applied sciences. Acknowledgement The authors wish to sincerely thank Covenant University (CUCRID) for the provision of enabling environment and finance. In addition, the authors thank the anonymous referees for their valuable contributions. Authors’ Contributions The concerned authors: GOA and SOE contributed significantly to this work. Both authors approved the final manuscript for publication. Ethics The authors declare that there exists no conflict of interest regarding this paper. References Aasaraai, A., 2011. Analytic solution for Newell-whitehead-segel equation by differential transform method. Middle East J. Scientiﬁc Res., 10: 270-273. Akinlabi, G.O. and S.O. Edeki, 2016. On approximate and closed-form solution method for initial-value wave-like models. Int. J. Pure Applied Math., 107: 449-456. DOI: 10.12732/ijpam.v107i2.14 Aksoy, Y. and M. Pakdemirli, 2010. New perturbation-iteration solutions for Bratu-type equations. Comput. Math. Applic., 59: 2802-2808.  DOI: 10.1016/j.camwa.2010.01.050 Edeki, S.O., G.O. Akinlabi and S.A. Adeosun, 2016. Analytic and numerical solutions of time-fractional linear schrödinger equation. Commun. Math. Applic., 7: 1-10.  Ezzati, R. and K. Shakibi, 2011. Using adomian’s decomposition and multiquadric quasi-interpolation methods for solving Newell-Whitehead equation. Proc. Comput. Sci., 3: 1043-1048.  DOI: 10.1016/j.procs.2010.12.171 He, J.H., 2005. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals, 26: 695-700.  DOI: 10.1016/j.chaos.2005.03.006 Khalid, M., M. Sultana, F. Zaidi and A. Uroosa, 2016. Solving linear and nonlinear Klein-Gordon equations by new perturbation iteration transform method. TWMS J. Applied Eng. Math., 6: 115-125.  Macías-Díaz, J. and J. Ruiz-Ramírez, 2011. A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell-Whitehead-Segel equation. Applied Numerical Math., 61: 630-640.  DOI: 10.1016/j.apnum.2010.12.008  Malomed, B.A., 1996. The Newell-Whitehead-Segel equation for travelling waves. Newell, A. and J. Whitehead, 1969. Finite bandwidth, finite amplitude convection. J. Fluid Mechan., 38: 279-303. DOI: 10.1017/S0022112069000176 Nourazar, S.S., M. Soori and A. Nazari-Golshan, 2011. On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method. Australian J. Basic Applied Sci. Fractals, 5: 1400-1411.  Patade, J. and S. Bhalekar, 2015. Approximate analytical solutions of Newell-Whitehead-Segel equation using a new iterative method. World J. Modell. Simulat., 11: 94-103.  Prakash, A. and M. Kumar, 2016. He’s variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation. J. Applied Anal. Comput., 6: 738-748. DOI: 10.11948/2016048 Saravanan, A. and N. Magesh, 2013. A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation. J. Egypt. Math. Society, 21: 259-265. DOI: 10.1016/j.joems.2013.03.004 Segel, L.A., 1969. Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mechan., 38: 203-224.  DOI: 10.1017/S0022112069000127 

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Perturbation Iteration Transform Method for the Solution of Newell-Whitehead-Segel Model Equations

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Perturbation Iteration Transform Method for the Solution of Newell-Whitehead-Segel Model Equations

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