Volume 67 | Issue 4 | Year 2021 | Article Id. IJMTT-V67I4P506 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I4P506

In this paper, the Tanh-coth and Banach contraction methods are proposed to solve the Burgers-Huxley and Kuramoto-Sivashinsky equations. The equations under study were first transformed into ordinary differential equations using specialized wave transformations as in Tanh-coth where solitary solutions were obtained, whereas the Banach contraction method gives an analytical solution after a finite number of iterations depending on the parameters. The result obtained showed, the methods are easy to implement, computationally less time consuming, accurate, reliable, promising and efficient.

[1] Wazwaz, A. M., Partial Differential Equations and Solitary Wave Theory, Springer Science & Business Media, (2010).

[2] Rajaraman, R., Solitons and Instantons, Elsevier Science Publishers, (1982).

[3] Shingareva, I.K., Carlos, L., Maple and Mathematics: A Problem-Solving Approach for Mathematics, Springer Science & Business media, (2011).

[4] Asokan, R., Vinodh, D., The Tanh-coth methpd for Soliton and Exact solutions of the Sawada-Kotera Equations, International Journal of Pure and Applied Mathematics, 117(13) (2017) 19-27.

[5] Gupta, A.K., Saha Ray, S., Numerical Treatment for the solution of fractional fifth-order Sawada-Kotera equations using second kind Chebyshev Wavelet method, Applied Mathematics Modelling, 39(17) (2015) 5121-5130.

[6] Darvishi, M.T., Khani, F., Numerical and Explicit solutions of the fifth-order Korteweg-de Vries Equations. Chaos, Solitons & Fractals, 39(5) (2009) 2484-2490.

[7] Wazwaz, A.M., The Hirota direct method and the Tanh-coth method for multiple-solitons of the Sawada-Kotera-Ito Seventh-order equations, Applied Mathematics and Computations, 199(1) (2008) 160-166

[8] Wazwaz, A.M., The Hirota Bilinear method and the Tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petriashvilli equation, Applied Mathematics Modelling, 200(1) (2008) 160-166.

[9] Lakshmanan, M., Tamizhmani, K.M., Painleve Analysis and Integrability Aspects of Nonlinear Evolution Equations, Solitons, Springer, Berlin, Heidelberg, (1988) 145-161.

[10] Fan, E., Auto-Backlund Transformation and Similarity reductions for generalized coefficient KDV equations, Physics of Letters A, 294(1) (2002) 26-30.

[11] Vaganan, B.M., Asokan, R., Direct Similarity analysis of generalized Burger’s equations and perturbation solutions of Euler-Painleve transcendent, Studies in Applied Mathematics, 111(4) (2003) 435-451.

[12] Manafian, J., Mehrdad, L., Bekir, A., Comparison Between the generalized Tanh-coth and the (G'/G) -expansion methods for solving NPDEs and NODEs. Pramana Journal of Physics, 87(6) (2016) 95.

[13] Fan, E., Hongqing, Z., A Note on the Homogenous Balance Method. Physics Letters A, 246(5) (1998) 403-406.

[14] Wazwaz, A.M., Abundant Solitons solutions for several forms of the fifth-order KDV equations by using the Tanh method, Applied Mathematics and Computations, 182(1) (2006) 283-300.

[15] Wazwaz, A.M., The extended Tanh method for new solitons solutions for many forms of the fifth-order KDV equations, Applied Mathematics and Computations, 84(2) (2007) 1002-1014.

[16] Gomez, C.A., Salas, A.H., The Variational Iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equations, Applied Mathematics and Computations, 217(4) (2010) 1408-1414.

[17] Wazwaz, A.M., The tanh-coth method for Solitons and Kink solutions for Nonlinear parabolic equations, Applied Mathematics and Computation, 188(2) (2007) 1467-1475.

[18] Parkes, E.J., Duffy, B.R., An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations. Computation in Physics Communication, 98(3) (1996) 288-300.

[19] Malfliet, W., Solitary Wave solutions of Nonlinear wave equations, American Journal of Physics, 60(7) (1992) 650-654.

[20] Nourazar, S.S., Mohsen, S. Nazari-Golshan, A., On the Exact solution of Burgers-Huxley Equation using the Homotopy Perturbation method, Journal of Applied Mathematics and Physics, 3 (2015) 285-294.

[21] Hashim, I., Norami, M.S.N., Said Al-Hadidi, M.R., Solving the generalized Burgers-Huxley Equation using the Adomian decomposition method, Mathematical and Computer Modelling 43 (2001) 1404-1411.

[22] Gao, H., and Zhao, R.X., New Exact Solutions to the Generalized Burgers-Huxley Equation, Applied Mathematics and Computation, 217 (2010) 1598-1603.

[23] Wang, X.Y., Zhu, Z.S., and Lu, Y.K., Solitary Wave solutions of the Generalized Burgers-Huxley Equations, Journal of Physics A: Mathematical and General, 23 (1990) 271.

[24] Darvishi, M.T., Kheybari, S., and Khani, F., Spectral Collocation Method and Darvishi’s Preconditioning to Solve the Generalized Burgers-Huxley Equation, Communication in Nonlinear Science and Numerical Simulation, 13 (2008) 2091-2103.

[25] Shivashinsky, G.I., Nonlinear Analysis of Hydrodynamic Instability in Laminar Flames, Derivation of Basic Equations, Acta Astronautica, 4(1) (1977) 1177-1206.

[26] Shivashinsky, G.I., Instabilities, pattern formation, and turbulence in flames, Annual review of Fluid Mechanics, 15 (1983) 179-199

[27] Kuramoto, Y., Diffusion-Induced Chaos in reaction Systems. Progression in Theortical Physics Supplement, 64 (1978) 346-367.

[28] Graham, W.G., William, E.S., Travelling Wave Analysis of Partial Differential Equations, Academic Press, Elsevier, USA, (2012).

[29] Abdel-Hamid, B., Exact Solutions of some Nonlinear evolution equations using symbolic computation. Computational Mathematics and Applications, 40 (2000) 291-302.

[30] Kudryashov, N.A., Exact Solution of the generalized Kuramoto-Shivashinsky equations, Physics Letters A, 147(5-6) (1990) 287-291.

[31] Wazwaz, A.M., An Analytic study of Compacton solution for variants of Kuramoto-Shivashinsky Equation, Applied Mathematics and Computation, 148 (2004) 571-585.

[32] Lai, H., Ma, C.F., Lattice Boltzmann method for the generalized Kuramoto-Shivashinsky equation, Physica A, 388 (2009) 1405-1412.

[33] Khater, A.H., Temsah, R.S., Numerical solutions of the generalized Kuramoto-Shivashinsky equation by Chebyshev Spectral collocation method, Computational Mathematics and Applications, 56 (2008) 1465-1472.

[34] Fan, E., Zhao, B., Extended tanh-function method and its applications to Nonlinear equations, Physics Letters A, 277 (2000) 212-218.

[35] Yan, X., Chi-Wang, S., Local discontinuous Galerkin methods for the Kuramoto-Shivashinsky equations and the Ito-type coupled KDV equations, Computational methods in Applied Mechanical Engineering, 195 (2006) 3430-3447.

[36] Soliman, A.A., A numerical simulation and explicit solutions of KDV-Burgers and Lax’s seventh-order KDV equations, Chaos, Solitons, Fractals, 29 (2006) 294-302.

[37] Chen, H., Zhang, H., New multiple soliton solutions to the general Burgers-Fisher equation and Kuramoto-Shivashinsky equations, Chaos, Solitons, Fractal, 19 (2004) 71-76.

[38] Tian-Shiang, Y., On Travelling wave solutions of the Kuramoto-Shivashinsky equations. Physica D, 110 (1997) 25-42.

[39] Baldwin, D., Goktas, O., Hereman, W., Hong, L., Martino, R.S., Miller, J.C., Symbolic computation of exact solution expressible in hyperbolic and elliptic function for Nonlinear PDEs. Journal of symbolic computation, 37 (2004) 669-705.

[40] Daftardar-Gejji, V., Jafari, H., An Iterative method or solving nonlinear functional equations. Journal of Mathematical Analysis and Applications, 316 (2006) 753-763.

[41] Daftardar-Gejji, V., Bhalekar, V. S., Solving nonlinear functional equation using Banach Contraction principle, Far East Journal of Applied Mathematics, 34(3) (2012) 303-314.

[42] Latif, A., Banach Contraction principle and its Generalizations, Springer International Publishing, Switzerland, (2014) 33-46.

[43] Kakde, R.V., Biradar, S.S., Hiremath, S.S., Solution of differential and integral equations using fixed point theorem, International Journal of Advanced Research in Computer Engineering and Technology, 3(5) (2014) 1656-1659.

Liberty Ebiwareme, "Banach Contraction Method and Tanh-coth Approach for the Solitary and Exact Solutions of Burger-Huxley and Kuramoto-Sivashinsky Equations.," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 67, no. 4, pp. 31-46, 2021. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V67I4P506