Hyperbolic trigonometric voyaging wave arrangement for non-integer adjusting term of Eckhaus condition and Klein Gordon condition

Abstract

The new wave solution of mathematical equations used in physics, engineering, and many applied sciences was found in this research using an alternative technique. For nonlinear partial differential equations without the term integer, our goal is to arrive at the analytical solution without the need for a new transformation to make the balancing term integer. To find the exact solutions to the Eckhaus equation and the cubic nonlinear Klein Gordon equation, as well as new type of complex hyperbolic trigonometric travelling wave solutions. In order to display the graphs showing the stationary wave, the parameters in these solutions are given specified values. Furthermore, few discussions about new complex solutions are presented. It is described by supplying the constants in traveling wave solutions, which are important both physically and mathematically, Finally, three-dimensional simulation is used to support these discussions.