The fundamental aim of the present article is to numerically solve the non-linear Equal-Width Wave (EW) equation. For this purpose, the nonlinear term appearing in the equation is first linearized by a Rubin-Graves-type approach. After that, to reduce the equation into a solvable discretized linear algebraic equation system, which is the essential part of this study, the Crank-Nicolson-type approximation and cubic Hermite collocation method are respectively applied to obtain the integration in the temporal and spatial domain directions. To demonstrate how good the offered method is at generating approximate numerical results, six experimental problems exhibiting different wave profiles, known as the motion of a single, interacting two and three, the Maxwellian initial, undular bore, and colliding soliton waves given with different initial and boundary conditions of the EW equation, will be taken into consideration and solved. Since only the first model problem has an exact solution among these solitary waves, to measure error magnitudes, the widely used mean squared and maximum norms between analytical and approximate solutions are calculated and also compared with those from other existing works available in the literature. Furthermore, the three conservation constants known as mass, moment, and energy quantities are also computed and presented throughout the wave simulations with increasing time. In addition, a tabular comparison of the newly computed norms and conservation constants shows that the current scheme produces better and more compatible solutions than those of most of the previous works with the same parameters. Apart from that, the stability analysis for this present scheme has been illustrated using the von Neumann method.
[1] H. A. Ali, A Biswas, and K. R. Raslan, Application of He’s Exp-function method and semi-inverse variational principle to equal width wave (EW) and modified equal width wave (MEW), Int. J. Phy. Sci., 7(7) (2012), 1035– 1043.
[2] A. H. A. Ali, Spectral method for solving the equal width equation based on Chebyshev polynomials, Nonlinear Dyn. 51 (2008), 59–70.
[3] S. Arora, R. Jain, and V. K. Kukreja, Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines, Appl. Math. Comput., 154 (2020), 1–16.
[4] S. Arora and I. Kaur, Applications of Quintic Hermite collocation with time discretization to singularly perturbed problems, Appl. Math. Comput., 316 (2018), 409–421.
[5] S. Arora, S. S. Dhaliwal, and V. K. Kukreja, Computationally efficient technique for weight functions and effect of orthogonal polynomials on the average, Appl. Math. Comput., 186 (2007), 623–631.
[6] J. Biazar and Z. Ayati, Application of the Exp-function method to the equal-width wave equation, Phys. Scr., 78 (2008), 045005 (4pp).
[7] I. Dağ and O. Ersoy, The exponential cubic B-spline algorithm for equal width equation, Adv. Studies Contemp. Math., 25(4) (2015), 525–535.
[8] İ, Dağ and B. Saka, A cubic B-spline collocatıon method for the EW equatıon, Math. Comput. Appl., 9(3) (2004), 381–392.
[9] Y. Dereli and R. Schaback, The Meshless Kernel-Based Method of Lines for solving the Equal Width Equation, Appl. Math. Comput., 219 (2013), 5224–5232.
[10] S. Dhawan, T. Ak, and G. Apaydın, Algorithms for numerical solution of the equal width wave equation using multi-quadric quasi-interpolation method, Int. J. Mod. Phys. C, 30 (2019).
[11] A. Dogan, Application of Galarkin’s method to equal width wave equation, Appl. Math. Comput., 160 (2005), 65–76.
[12] A. Esen, A numerical solution of the equal width wave equation by a lumped Galarkin method, Appl. Math. Comput., 168 (2005), 270–282.
[13] A. Esen and S. Kutluay, A linearized implicit finite-difference method for solving the equal width wave equation, Int. J. Comput. Math., 83(3) (2006), 319–330.
[14] H. Fazal-i, A. Inayet, and A. Shakeel, Septic B-spline Collocation method for numerical solution of the Equal Width Wave (EW) equation, Life Science Journal, 10 (2013), 253–260.
[15] I. A. Ganaie, S. Arora, and V. K. Kukreja, Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation, Int. J. Comput. Math., 93(1) (2016), 223–235.
[16] I. A. Ganaie, B. Gupta, N. Parumasur, P. Singh, and V. K. Kukreja, Asymptotic convergence of cubic Hermite collocation method for parabolic partial differential equation, Appl. Math. Comput., 220 (2013), 560–567.
[17] L. R. T. Gardner, G. A. Gardner, F. A. Ayoup, and N. K. Amein, Simulations of the EW undular bore, Commun. Numer. Methods Eng., 13 (1997), 583–592.
[18] I. A. Ganaie and V. K. Kukreja, Numerical solution of Burgers’ equation by cubic Hermite collocation method, Appl. Math. Comput., 237 (2014) 571–581.
[19] A. Ghafoor and S. Haq, An efficient numerical scheme for the study of equal width equation, Results Phys., 9 (2018), 1411-1416.
[20] B. İnan and A. R. Bahadır, A numerical solution of the equal width wave equation using a fully implicit finite difference method, Turkish J. Math. and Comp. Sci., (2014), Article ID 20140037, 1-14.
[21] S. P. Kaur, A. K. Mittal, V. K. Kukreja, A. Kaundal, N. Parumasur, and P. Singh, Analysis of a linear and non-linear model for diffusion–dispersion phenomena of pulp washing by using quintic Hermite interpolation polynomials, Afrika Matematika, 32 (2021), 997-1019.
[22] S. P. Kaur, A. K. Mittal, V. K. Kukreja, N. Parumasur, and P. Singh, An efficient technique for solution of linear and nonlinear diffusion-dispersion models, AIP Conference Proceedings, 2018.
[23] N. A. Kudryashov, Generalized Hermite polynomials for the Burgers hierarchy and point vortices, Chaos, Solitons Fractals, 151 (2021), 111256.
[24] A. Kumari and V. K. Kukreja, Robust septic Hermite collocation technique for singularly perturbed generalized Hodgkin–Huxley equation, Int. J. Comput. Math., 2021.
[25] A. Kumari and V. K. Kukreja, Septic Hermite collocation method for the numerical solution of Benjamin–Bona– Mahony–Burgers equation, J. Differ. Equations Appl., 27 (2021), 1193–1217.
[26] S. Kutluay, N. M. Yağmurlu, and A. S. Karakaş, An Effective Numerical Approach Based on Cubic Hermite B- spline Collocation Method for Solving the 1D Heat Conduction Equation, New Trends in Math. Sci., 10(4) (2022), 20–31.
[27] M. Lakestani and M. Dehghan, Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions, Appl. Math. Modell., 36(2) (2012), 605–617.
[28] M. Lakestani, Numerical solutions of the KdV equation using B-spline functions, ran. J. Sci. Technol. Trans. Electr. Eng., Transactions A: Science, 41 (2017), 409–417.
[29] D. Lu, A. R. Seadawy, and A. Ali, Dispersive traveling wave solutions of the Equal-Width and Modified Equal- Width equations via mathematical methods and its applications, Results Phys., 9 (2018), 313–320.
[30] G. T. Lubo and G. F. Duressa, Linear B-spline finite element Method for solving delay reaction diffusion equation Comput. Methods Differ. Equ., 11(1) (2023), 161–174.
[31] A. K. Mittal, I. A. Ganaie, V. K. Kukreja, N. Parumasur, and P. Singh, Solution of diffusion–dispersion models using a computationally efficient technique of orthogonal collocation on finite elements with cubic Hermite as basis, Comp. and Chem. Eng., 58 (2013), 203–210.
[32] P. J. Morrison, J. D. Meiss, and J. R. Cary, Scattering of Regularized-Long-Wave solitary waves, Physica 11D, (1984), 324–336.
[33] A. H. Msmali, M. Tamsir, and A. A. H. Ahmadini, Unified and extended trigonometric B-spline DQM for the numerical treatment of three-dimensional wave equations, Ain Shams Eng. J., 15(2) (2024), 102382.
[34] R. I. Nuruddeen, K. S. Aboodh, and K. K. Ali, Investigating the tangent dispersive solitary wave solutions to the Equal Width and Regularized Long Wave equations, J. King Saud Univ. Sci.
[35] P. J. Olver, Euler operators and conservation laws of the BBM equation, Math Proc. Camb. Phil. Soc. 85 (1979), 143–160.
[36] J. I. Ramos, Explicit finite difference methods for the EW and RLW equations, Appl. Math. Comput., 179 (2006), 622–638.
[37] K. R. Raslan, A computational method for the equal width equation, Int. J. Comput. Math., 81 (2004), 63–72.
[38] K. R. Raslan, Collocation method using quartic B-spline for the equal width (EW) equation, Appl. Math. Comput., 168 (2005), 795–805.
[39] T. Roshan, A Petrov-Galerkin Method for Equal width equation, Appl. Math. Comput., 218 (2011), 2730–2739.
[40] S. G. Rubin and R. A. Graves, A cubic spline approximation for problems in fluid mechanics, National aeronautics and space administration, Technical Report, Washington, 1975.
[41] B. Saka, I. Dağ, Y. Dereli, and A. Korkmaz, Three different methods for numerical solutions of the EW equation, Eng. Anal. Boundary Elem., 32 (2008), 556-566.
[42] B. Nemati Saray, M. Lakestani, and C. Cattani, Evaluation of mixed Crank–Nicolson scheme and Tau method for the solution of Klein–Gordon equation Appl. Math. Comput., 331 (2018), 169–181.
[43] M. Uddin, RBF-PS scheme for solving the equal width equation, Appl. Math. Comput., 222 (2013), 619–631.
[44] N. M. Ya˘gmurlu and A. S. Karaka¸s, Numerical solutions of the equal width equation by trigonometric cubic B- spline collocation method based on Rubin–Graves type linearization, Numer. Methods Partial Differ. Equations, 36 (2020), 1170–1183.
[45] A. Yousaf, T. Abdeljawad, M. Yaseen, and M. Abbas, Novel Cubic Trigonometric B-Spline Approach Based on the Hermite Formula for Solving the Convection-Diffusion Equation, Math. Problems in Eng., (2020).
[46] S. I. Zaki, A least-squares finite element scheme for the EW equation, Comp. Meth. Appl. in Mech. Eng., 189 (2000), 587–594.
Kutluay, S. , Yağmurlu, N. Murat and Karakaş, A. Sercan (2026). A new perspective for simulations of the equal-width wave equation. Computational Methods for Differential Equations, 14(3), 1112-1129. doi: 10.22034/cmde.2025.62400.2747
MLA
Kutluay, S. , , Yağmurlu, N. Murat, and Karakaş, A. Sercan. "A new perspective for simulations of the equal-width wave equation", Computational Methods for Differential Equations, 14, 3, 2026, 1112-1129. doi: 10.22034/cmde.2025.62400.2747
HARVARD
Kutluay, S., Yağmurlu, N. Murat, Karakaş, A. Sercan (2026). 'A new perspective for simulations of the equal-width wave equation', Computational Methods for Differential Equations, 14(3), pp. 1112-1129. doi: 10.22034/cmde.2025.62400.2747
CHICAGO
S. Kutluay , N. Murat Yağmurlu and A. Sercan Karakaş, "A new perspective for simulations of the equal-width wave equation," Computational Methods for Differential Equations, 14 3 (2026): 1112-1129, doi: 10.22034/cmde.2025.62400.2747
VANCOUVER
Kutluay, S., Yağmurlu, N. Murat, Karakaş, A. Sercan A new perspective for simulations of the equal-width wave equation. Computational Methods for Differential Equations, 2026; 14(3): 1112-1129. doi: 10.22034/cmde.2025.62400.2747
)