Innovative approache for the nonlinear atangana conformable Klein-Gordon equation unveiling traveling wave patterns

The aim of current work is to establish novel traveling wave solutions of the nonlinear Atangana conformable Klein - Gordon equation using a new extended direct algebraic technique. The Klein - Gordon equation is the relativistic state of the Schrödinger equation with a second - order time derivative and zero spin. Complex wave variable transformation is used to convert Atangana conformable nonlinear differential equation into an ordinary differential equation. Using the proposed technique based on Maple software structure, various types of solutions, such as, generalized trigonometric, generalized hyperbolic, and exponential functions, are established. When special parameteric values are considered for this method, solitary wave solutions can be obtained through other methods, such as the ($\frac{G^{\prime }}{G}$)-expansion method, the modified Kudryashov method, the sub-equation method, and so forth. A physical explanation is provided for the solutions under consideration to enhance comprehension of the physical phenomena resulting from the obtained solutions, provided that the physical parameters are set appropriately using 3D, 2D, and contour simulations. The results demonstrated that the new extended direct algebraic method provides a more potent mathematical tool for solving numerous more nonlinear partial differential equations with the aid of symbolic computation.