Main Article Content
This paper aims to propose a hybrid approach of two powerful methods, namely the differential transform and finite difference methods, to obtain the solution of the coupled Whitham-Broer-Kaup-Like equations which arises in shallow-water wave theory. The capability of the method to such problems is verified by taking different parameters and initial conditions. The numerical simulations are depicted in 2D and 3D graphs. It is shown that the used approach returns accurate solutions for this type of problems in comparison with the analytic ones.
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Wang ML, Zhou Y, Li Z. Application of a homogenous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A, 1996; 216: 67-75.
Song M, Cao J, Guan X. Application of the bifurcation method to the Whitham–Broer–Kaup-Like equations. Math. and Comput. Model. 2012 February; 55: 688–696.
Whitham GB, Variation methods and applications to water waves. Proc. of the R. Soc. Lond. 1967; Series A 299: 6-25.
Broer LJF. Approximate equations for long water waves, Appl. Sci. Res. 1975; 31: 377-395.
Kaup DJ. A higher-order water wave equation and the method for solving it. Prog. Theor. Phys. 1975; 54: 396-408.
Kupershmidt BA. Mathematics of dispersive water waves. Comm. Math. Phys. 1985; 99: 51–73.
Nikkar A, Ahmadiasl R. A Novel Method For Solving Nonlinear Whitham-Broer-Kaup Equation System. Proceedings of Academics World 5th International Conference, Paris, France. 2015; 6-9.
Ahmad J, Mushtaq M, Sajjad N. Exact solution of Whitham–Broer–Kaup shallow water wave equation. J. Sci. Arts. 2015; 1(30): 5-12.
Wang XB,Tian SF, Qin CY, Zhang TT. Lie Symmetry Analysis, Analytical solutions, and conservation laws of the generalised Whitham–Broer–Kaup–Like equations. Zeitschriftfür Naturforschung A. 2017; 72(3): 269–279.
Chu HP, Chen CL. Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem. CNSNS. 2008; 13: 1605–1614.
Maerefat M, Rad MT, Ghazizadeh HR . Hybrid differential transform-finite difference solution of 2D transient nonlinear annular fin equation. Iranian J. Mech. Eng. 2010; 11: No.2.
Süngü IC, Demir H. Application of the hybrid differential transform method to the nonlinear equations. Appl. Math. 2012; 3: 246-250.
Chu SP. Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem. Whampoa - An Interdisciplinary J. 2014; 66: 15-26.
Mosayebidorcheh S, Sheikholeslami M, Hatami M , Ganji DD. Analysis of turbulent MHD couettenanofluid flow and heat transfer using hybrid DTM-FDM. Particuology. 2016 Jun 1;26:95-101.
Kazem S, Dehghan M. Application of finite difference method of lines on the heat equation. Numerical Methods for Partial Differential Equations . 2018 Mar; 34(2):626-660.
Ahmed S. Finite difference method for solving the initial value ordinary differential equations (ODE). IJRASET.2018; 6(1):2545-2549.
Yazengaw N. Convergence analysis of finite difference method for differential equation. J. Phys. Math. 2017;8(3)1-3.
Raslan KR, Biswas A, Abu Sheer ZF. Differential transform method for solving partial differential equations with variable coefficients,. IJPS. 2012; 7(9): 1412-1419.
Odibat ZM, Kumar S, Shawagfeh N, Alsaedi A, Hayat T. A study on the convergence conditions of generalized differential transform method. Math. Meth. Appl. Sci. 2017; 40: 40-48.
Hatami M, Ganji DD, Sheikholeslami M. Differential transformation method for mechanical engineering problems. Academic Press of Elsevier, 2017.