A new numerical approach to the solution of the nonlinear Kawahara equation by using combined Taylor–Dickson approximation

Document Type : Research Article


Department of Mathematics, School of Applied Sciences, KIIT University, Odisha-751024, India.


This article presents a novel numerical approach to the solution of the nonlinear Kawahara equation. The desired approximations are obtained from the combination of Dickson polynomials and Taylor’s expansion. The combined approach is based on Taylor’s expansion for discretizing the time derivative and Dickson polynomials for space derivatives. The problem will be converted into a system of linear algebraic equations for each time step via some suitable collocation points. Error estimation is presented after obtaining the approximate solution. The newly proposed technique is compared with some existing numerical methods to show the method’s applicability, accuracy, and efficacy. Two problems are solved to demon-strate the method’s power and effect, and the results are presented as a table and graphics.


Main Subjects

[1] Abbasbandy, S. Homotopy analysis method for the Kawahara equation, Nonlinear Anal. Real World Appl. 11(1) (2010), 307–312.
[2] Ak, T. and Karakoc, S.B.G. A numerical technique based on collocation method for solving modified Kawahara equation, J. Ocean Eng. Sci. 3(1) (2018), 67–75.
[3] Alipour, M. and Soradi Zeid, S. Optimal control of Volterra integro-differential equations based on interpolation polynomials and collocation method, Comput. Methods Differ. Equ. 11(1) (2023), 52–64.
[4] Aljahdaly, N.H. and Alweldi, A.M. On the modified Laplace homotopy perturbation method for solving damped modified kawahara equation and its application in a fluid, Symmetry, 15(2) (2023), 394.
[5] Başhan, A., Uçar, Y., Yağmurlu, N.M. and Esen, A., 2016, October. Numerical solution of the complex modified Korteweg-de Vries equation by DQM, In Journal of Physics: Conference Series (Vol. 766, No. 1, p. 012028). IOP Publishing.
[6] Başhan, A., Yağmurlu, N.M., Ucar, Y. and Esen, A., 2021. Finite differ-ence method combined with differential quadrature method for numerical computation of the modified equal width wave equation, Numer. Methods Partial Differential Equations 37(1) (2021), 690–706.
[7] Bhargava, M. and Zieve, M.E. Factoring Dickson polynomials over finite fields, Finite Fields and Their Appl. 5(2) (1999), 103–111.
[8] Bhatter, S., Mathur, A., Kumar, D., Nisar, K.S. and Singh, J. Fractional modified Kawahara equation with Mittag–Leffler law, Chaos, Solitons Fractals, 131 (2020), 109508.
[9] Biswas, A. Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett. 22(2) (2009), 208–210.
[10] Brewer, B.W., On certain character sums, Trans. Amer. Math. Soc. 99(2) (1961), 241–245.
[11] Dickson, L.E., The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. of Math. 11(1-6) (1896/97), 65–120.
[12] Fernando, N., A study of permutation polynomials over finite fields, Thesis (Ph.D.)–University of South Florida. ProQuest LLC, Ann Arbor, MI, 2013. 109 pp.
[13] Haq, S. and Uddin, M. RBFs approximation method for Kawahara equa-tion, Eng. Anal. Bound. Elem. 35(3) (2011), 575–580.
[14] He, D., New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity, Nonlinear Dynam. 82 (2015), 1177–1190.
[15] Karakoç, S.B.G., Zeybek, H. and Ak, T. Numerical solutions of the Kawahara equation by the septic B spline collocation method, Stat. Op-tim. Inf. Comput. 2(3) (2014), 211–221.
[16] Kaya, D. and Al-Khaled, K. A numerical comparison of a Kawahara equation, Phys. Lett. A, 363 (5-6) (2007), 433–439.
[17] Kreyszig, E. Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978.
[18] Kürkçü, Ö.K., Aslan, E. and Sezer, M. A numerical approach with error estimation to solve general integro-differential-difference equations using Dickson polynomials, Appl. Math. Comput 276 (2016), 324–339.
[19] Kürkçü, Ö.K., Aslan, E. and Sezer, M. A novel collocation method based on residual error analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials, Sains Malays, 46 (2017), 335–347.
[20] Kürkçü, Ö.K., Aslan, E., Sezer, M. and İlhan, Ö. A numerical approach technique for solving generalized delay integro-differential equations with functional bounds by means of Dickson polynomials, Int. J. Comput. Methods, 15(05) (2018), 1850039.
[21] Lahiji, M.A. and Aziz, Z.A. Numerical solution for Kawahara equation by using spectral methods, IERI Procedia, 10 (2014), 259–265.
[22] Nikan, O. and Avazzadeh, Z. A locally stabilized radial basis function partition of unity technique for the sine-Gordon system in nonlinear optics, Math. Comput. Simul. 199 (2022), 394–413.
[23] Nikan, O., Avazzadeh, Z., Machado, J.T. and Rasoulizadeh, M.N., An accurate localized meshfree collocation technique for the telegraph equa-tion in propagation of electrical signals, Eng. Comput. 39(3) (2023), 2327–2344.
[24] Nikan, O., Avazzadeh, Z. and Rasoulizadeh, M.N., Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory, Nonlinear Dynam. 106(1) (2021), 783–813.
[25] Nikan, O., Avazzadeh, Z. and Rasoulizadeh, M.N., Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaces, Eng. Anal. Bound. Elem. 143 (2022), 14–27.
[26] Özer, S. Numerical solution by quintic B-spline collocation finite el-ement method of generalized Rosenau–Kawahara equation, Math. Sci. 16(3) (2022), 213–224.
[27] Polat, N., Kaya, D. and Tutalar, H.I. A analytic and numerical solu-tion to a modified Kawahara equation and a convergence analysis of the method, Appl. Math. Comput. 179(2) ( 2006), 466–472.
[28] Rasoulizadeh, M.N. and Rashidinia, J. Numerical solution for the Kawa-hara equation using local RBF-FD meshless method, J. King Saud Univ. Sci. 32(4) (2020), 2277–2283.
[29] Sahu, P.K., Ray, S.S. Numerical solutions for Volterra integro-differential forms of Lane-Emden equations of first and second kind using Legendre multiwavelets, Electron. J. Differ. Equ. 2015 (28) (2015), 1–11.
[30] Sahu, P.K. and Ray, S.S., Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system, Appl. Math. Comput. 256 (2015), 715–723.
[31] Saldır, O., Sakar, M.G. and Erdogan, F., Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate, Comput. Appl. Math. 38 (2019), 1–23.
[32] Soltanalizadeh, B., Application of differential transformation method for numerical analysis of Kawahara equation, Aust. J. Basic Appl. Sci. 5(12) (2011), 490–495.
[33] Wang, X. and Cheng, H. Solitary wave solution and a linear mass-conservative difference scheme for the generalized Korteweg–de Vries–Kawahara equation, Comput. Appl. Math. 40 (2021), 1–26.
[34] Zara, A., Rehman, S.U., Ahmad, F., Kouser, S. and Pervaiz, A. Numeri-cal approximation of modified Kawahara equation using Kernel smoothing method, Math. Comput. Simul. 194 (2022), 169–184.