An Exact Analytical Solution to the Shallow Water Equations Near Beaches

M. Yourdkhani, Saber Zarrinkamar


The nonlinear differential equation governing the dynamics of water waves can be well approximated by a linear counterpart in the case of shallow waters near beaches. The linear equation, which is of second order nature, cannot be exactly solved in many apparently simple cases. In our work, we consider the shape of system as a complete second-order polynomial which contains the constant (step-like), linear and quadratic shapes near the beach. We then apply some novel transformations and transform the problem into a form which can be solved in an exact analytical manner via the powerful Nikiforov-Uvarov technique. The eigenfunctions of the problem are obtained in terms of the Jacobi polynomials and the eigenvalue equation is reported for any arbitrary mode. 

Full Text:



Th. Gallouet, J. –M. Herard and N. Seguin, Computers & Fluids 32 (2003) 479.

P. Garcia-Navarro and M. E. Vazquez-Cendon. Comput Fluids 29 (2000) 951.

V. Aizinger and C. Dawson, Comp. Meth. App. Mech. Eng. 196 (2007) 734.

J. C. Butcher, App. Num. Math. 20 (1996) 247.

O. Gourgue et al., Adv. Water Res. 32 (2009) 1726.

C. B. Vreugdenhil, C. B. (1994). Numerical methods for shallow-water waves, volume 13 of Water Science and Technology Library. Kluwer Academic Publishers.

Shi-Hai Dong, Factorization methods in quantum mechanics, Springer-Verlag, New-York, 2007.

Shi-Hai Dong, Wave equations in higher dimensions, Springer-Verlag, New-York, 2011.

P. J. Olver. Applications of Lie groups to differential equations, Springer-Verlag, New York, 1986.

P. L.-F. Liu, P. Lynett and C. E. Synolakis, J. Fluid Mech. (2003), vol. 478, pp. 101–109.

A. F. Nikiforov and V. B.Uvarov, Special Functions of Mathematical Physics (Birkhauser, Basel, 1988).

C. Tezcan and R. Sever, Int. J. Theor. Phys. 48 (2009) 337.

A. D. Antia, I. O. Akpan, A. N. Ikot, E. Maghsoodi, S. Zarrinkamar, H. Hassanabadi, Annales Universitatis Mariae Curie-Sklodowska, sectio AAA–Physica 70 (2015) 167.

Ch. C. Mei, The Applied Dynamics of Ocean Surface Waves, John Wiley & Sons, New York, 1983.

Data publikacji: 2017-02-23 09:57:46
Data złożenia artykułu: 2016-07-21 06:58:10


  • There are currently no refbacks.

Copyright (c) 2017 saber zarrinkamar

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.