On the Extended Tanh Method Applications of Nonlinear Equations

Nonlinear evolution equations have a major role in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, solid state physics, chemical kinematics, chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In recent years, quite a few methods for obtaining explicit traveling and solitary wave solutions of nonlinear evolution equations have been proposed. A variety of powerful methods, such as inverse scattering method [1, 16], bilinear tranformation [9, 11, 15], the tanhsech method [12, 14, 21], extended tanh method [4, 6, 23], sine-cosine method [24–26, 30], F-expansion method [3], and homogeneous balance method [5, 17, 29] were used to investigate nonlinear dispersive and dissipative problems. The pioneer work Malfiet in [12, 13] introduced the powerful tanh method for a reliable treatment of the nonlinear wave equations. The useful tanh method is widely used by many work such as in [18–20, 23] and by the references therein. Later, the extended tanh method, developed by Wazwaz [27, 28], is a direct and effective algebraic method for handling nonlinear equations. Various extensions of the method were developed as well. Our first interest in present work being in implementing the extended tanh method to stress its power in handling nonlinear equations so that one can apply it to models of various types of nonlinearity. The next interest is in the determination of exact travelling wave solutions for Benjamin-Bona-Mahony equation, modified Benjamin-Bona-Mahony equation and KdV-Burger equation. Searching for exact solutions of nonlinear problems has attracted a considerable amount of research work where computer symbolic systems facilitate the computational work.