Rostam K. Saeed

1 Department of Mathematics, College of Science, Salahaddin University, Kurdistan Region, Iraq.

2 Department of Mathematics, College of Education, Duhok University, Kurdistan Region, Iraq.

In this paper, we presented finite difference methods for solving nonlinear Whitham-Broer-Kaup (WBK) shallow water model numerically. We first subdivided the domain of the model by a net with a finite number of mesh points, and the derivative at each point replaced by explicit, Crank-Nicolson, and exponential finite difference approximations. The result is the system of algebraic equations which when solved, provide an approximation to the solutions of WBK model at the selected grid points. Also, a comparison has been made between the approximate solutions obtained by the proposed methods and the exact solutions. Numerical results represented in tables and figures with the help of MATLAB R2015a.

Finite difference method,

Whitham-Broer-Kaup shallow water model.

[1] Ablowitz, M. J. and Clarkson, P. A., "Solitons, Nonlinear Evolution Equations and Inverse Scattering". Cambridge University Press, (1991).

[3] Bhattacharya, M. C., "A new improved finite difference equation for heat transfer during transient change". Applied Mathematical Modelling, Vol. 10, No.1, pp. 68-70. (1986).

[4] Duffy, D. J.," Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach". John Wiley & Sons, (2006).

[5] Engui, F. and Hongqing, Z., "Backlund transformation and exact solutions for Whitham-Broer-kaup equations in shallow water". Applied Mathematics and Mechanics, Vol. 19, No. 8, pp.713-716. (1998).

[6] Handschuh, R. F., "An Exponential Finite Difference Technique for Solving Partial Differential Equations". Master of Science Thesis, University of Toledo, (1987).

[7] Kupershmidt, B. A., "Mathematics of dispersive water waves". Communications in Mathematical Physics, Vol. 99, No. 1, pp.51-73. (1985).

[8] Lapidus, L. and Pinder, G. F.,"Numerical Solution of Partial Differential Equations in Science and Engineering". John Wiley & Sons, (1999).

[9] Logan, J. D., "Applied Mathematics: A Contemporary Approach". John Wiley & Sons, (1987).

[10] Smith, G. D., "Numerical Solution of Partial Differential Equations: Finite Difference Methods". Oxford University Press, (1985).

[11] Mishra, L. N., "On Existence and Behavior of Solutions to Some Nonlinear Integral Eequations with Applications". Ph.D. Thesis, National Institute of Technology, Silchar 788 010, Assam, India, (2016),

[12] Mishra, L. N.; Agarwal, R. P. and Sen, M., "Solvability and asymptotic behavior for some nonlinear quadratic integral equation involving Erdèlyi-Kober fractional integrals on the unbounded interval". Progress in Fractional Differentiation and Applications, Vol. 2, No. 3, pp.153-168. (2016).

[13] Mishra, L. N., Sen, M., "On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order". Applied Mathematics and Computation, Vol. 285, pp.174-183. (2016).

[14] Mishra, L. N.; Sen, M. and Mohapatra, R. N., "On existence theorems for some generalized nonlinear functional-integral equations with applications". Filomat, accepted on March 21, 2016, in press.

[15] Mishra, L. N.; Srivastava, H. M. and Sen, M., "On existence results for some nonlinear functional-integral equations in Banach algebra with applications". Int. J. Anal. Appl., Vol. 11, No. 1, pp.1-10. (2016).

[16] H. S. Shukla, H. S.; Tamsir, M.; Srivastava, V. K. and Rashidi, M. M., "Modified cubic B-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burger equation", Modern Physics Letters B, Vol. 30, No. 11, pp. 1-17. (2016).

[17] Suarez, P. U. and Morales J. H., "Numerical solutions of two-way propagation of nonlinear dispersive waves using radial basis functions". International Journal of Partial Differential Equations, Vol. (2014), pp.1-8. (2014).

[18] Thakumar, M. S., "Computer Based Numerical Analysis". Khanna Publishers, (1989).

[19] Wang, L.; Gao, Y. T.; Gai, X. L.; Yu, X.; and Sun, Z. Y., "Vadermonde-type odd-soliton solutions for the Whitham–Broer–Kaup model in the shallow water small amplitude regime". Journal of Nonlinear Mathematical Physics, Vol. 17, No. 2, pp.197-211, (2010).

[20] Wani, S. S. and Thakar, S.H., "Crank-Nicolson type method for Burgers equation". International Journal of Applied Physics and Mathematics, Vol.3, No. 5, pp.324-328. (2013).

[21] Xu, T.; Li, J.; Zhang, H-Q.; Zhang, Y-X.; Yao, Z-Z. and Tian, B., New extension of the tanh-function method and application to the Whitham-Broer-Kaup shallow water model with symbolic computation. Physics Letters A, Vol. 369, No. 5, pp.458-463. (2007).

^{1}, Mohammed I. Sadeeq^{2}1 Department of Mathematics, College of Science, Salahaddin University, Kurdistan Region, Iraq.

2 Department of Mathematics, College of Education, Duhok University, Kurdistan Region, Iraq.

Original: 29 May 2016, Revised: 15 September 2016, Accepted: 16 October 2016, Published online: 20 March 2017

**DOI:**

**https://doi.org/10.17656/jzs.10597**

**Abstract**

In this paper, we presented finite difference methods for solving nonlinear Whitham-Broer-Kaup (WBK) shallow water model numerically. We first subdivided the domain of the model by a net with a finite number of mesh points, and the derivative at each point replaced by explicit, Crank-Nicolson, and exponential finite difference approximations. The result is the system of algebraic equations which when solved, provide an approximation to the solutions of WBK model at the selected grid points. Also, a comparison has been made between the approximate solutions obtained by the proposed methods and the exact solutions. Numerical results represented in tables and figures with the help of MATLAB R2015a.

**Key Words:**

Finite difference method,

Whitham-Broer-Kaup shallow water model.

**References**

[1] Ablowitz, M. J. and Clarkson, P. A., "Solitons, Nonlinear Evolution Equations and Inverse Scattering". Cambridge University Press, (1991).

[2] Bahadir, A. R., "Exponential finite-difference method applied to Kortewge-de Vries equation for small times". Applied Mathematics and Computation, Vol. 160, No. 3, pp. 675- 682. (2005).

[3] Bhattacharya, M. C., "A new improved finite difference equation for heat transfer during transient change". Applied Mathematical Modelling, Vol. 10, No.1, pp. 68-70. (1986).

[4] Duffy, D. J.," Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach". John Wiley & Sons, (2006).

[5] Engui, F. and Hongqing, Z., "Backlund transformation and exact solutions for Whitham-Broer-kaup equations in shallow water". Applied Mathematics and Mechanics, Vol. 19, No. 8, pp.713-716. (1998).

[6] Handschuh, R. F., "An Exponential Finite Difference Technique for Solving Partial Differential Equations". Master of Science Thesis, University of Toledo, (1987).

[7] Kupershmidt, B. A., "Mathematics of dispersive water waves". Communications in Mathematical Physics, Vol. 99, No. 1, pp.51-73. (1985).

[8] Lapidus, L. and Pinder, G. F.,"Numerical Solution of Partial Differential Equations in Science and Engineering". John Wiley & Sons, (1999).

[9] Logan, J. D., "Applied Mathematics: A Contemporary Approach". John Wiley & Sons, (1987).

[10] Smith, G. D., "Numerical Solution of Partial Differential Equations: Finite Difference Methods". Oxford University Press, (1985).

[11] Mishra, L. N., "On Existence and Behavior of Solutions to Some Nonlinear Integral Eequations with Applications". Ph.D. Thesis, National Institute of Technology, Silchar 788 010, Assam, India, (2016),

[12] Mishra, L. N.; Agarwal, R. P. and Sen, M., "Solvability and asymptotic behavior for some nonlinear quadratic integral equation involving Erdèlyi-Kober fractional integrals on the unbounded interval". Progress in Fractional Differentiation and Applications, Vol. 2, No. 3, pp.153-168. (2016).

[13] Mishra, L. N., Sen, M., "On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order". Applied Mathematics and Computation, Vol. 285, pp.174-183. (2016).

[14] Mishra, L. N.; Sen, M. and Mohapatra, R. N., "On existence theorems for some generalized nonlinear functional-integral equations with applications". Filomat, accepted on March 21, 2016, in press.

[15] Mishra, L. N.; Srivastava, H. M. and Sen, M., "On existence results for some nonlinear functional-integral equations in Banach algebra with applications". Int. J. Anal. Appl., Vol. 11, No. 1, pp.1-10. (2016).

[16] H. S. Shukla, H. S.; Tamsir, M.; Srivastava, V. K. and Rashidi, M. M., "Modified cubic B-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burger equation", Modern Physics Letters B, Vol. 30, No. 11, pp. 1-17. (2016).

[17] Suarez, P. U. and Morales J. H., "Numerical solutions of two-way propagation of nonlinear dispersive waves using radial basis functions". International Journal of Partial Differential Equations, Vol. (2014), pp.1-8. (2014).

[18] Thakumar, M. S., "Computer Based Numerical Analysis". Khanna Publishers, (1989).

[19] Wang, L.; Gao, Y. T.; Gai, X. L.; Yu, X.; and Sun, Z. Y., "Vadermonde-type odd-soliton solutions for the Whitham–Broer–Kaup model in the shallow water small amplitude regime". Journal of Nonlinear Mathematical Physics, Vol. 17, No. 2, pp.197-211, (2010).

[20] Wani, S. S. and Thakar, S.H., "Crank-Nicolson type method for Burgers equation". International Journal of Applied Physics and Mathematics, Vol.3, No. 5, pp.324-328. (2013).

[21] Xu, T.; Li, J.; Zhang, H-Q.; Zhang, Y-X.; Yao, Z-Z. and Tian, B., New extension of the tanh-function method and application to the Whitham-Broer-Kaup shallow water model with symbolic computation. Physics Letters A, Vol. 369, No. 5, pp.458-463. (2007).