Abstract
The tanh method is used to compute travelling waves solutions of one-dimensional nonlinear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. In this article, we introduce a new general form of tanh transformation and solve well-known nonlinear partial differential equations in which tanh method becomes weaker in the sense of obtaining general form of solutions.
Highlights
Nonlinear partial differential equations are encountered in various fields of mathematics, physics, chemistry, and mathematical biology, and numerous applications
We introduce a new technique, by constructing a new model which is the general form of tanh function, whereby we obtain exact solutions in case of the nonlinear PDEs
We see that the solution depends on travelling frame ξ, and η which characterizes the travelling wave solution
Summary
Nonlinear partial differential equations are encountered in various fields of mathematics, physics (fluid dynamics [14]), chemistry, and mathematical biology (population dynamics [11]), and numerous applications. Exact (closed–form) solutions of differential equations play a crucial role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Because of the increasing interest in finding exact solutions for those problems, many powerful methods have been developed, such as, tanh-function method (see, e.g, [5, 7, 8, 9, 10]), inverse scattering method (see, e.g, [1]) and direct algebraic method (see, e.g, [2, 3, 4]). We investigate the nonlinear wave and evolution equations (one dimensional) which are written as. A(u, ut, ux, uxx, ...) = 0 or A(u, utt, ux, uxx, ...) = 0 We assume that these equations admit exact travelling wave solutions. We assume that the wave number is greater than zero. Nonlinear partial differential equations, exact solutions, nonlinear travelling waves.
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