The Klein–Fock–Gordon and Tzitzeica dynamical equations with advanced analytical wave solutions

In this manuscript, two mathematical approaches have been functionalized to discover novel wave results of 3rd-order Klein–Gordon and Tzitzeica equations. With the alliance of Mathematica, the competency of these methods for discovering these exact solutions have been more exhibited. As a result, several solitary solutions are constructed and indicated by hyperbolic solutions, diverse combinations of trigonometric and exponential results. Furthermore, employed techniques are more efficient techniques for exploring essential nonlinear waves that enhance a variety of dynamic models that arises in nonlinear fields. All drafting is given out to express the properties of the innovative explicit analytic solutions. Hence our proposed schemes are directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also.