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This manuscript handles the nonlinear Klein–Fock–Gordon ( KFG ) equation by applying two recent computational schemes (generalized exponential function (GEF) and generalized Riccati expansion (GRE) methods) to construct abundant novel wave solutions The considered model is the generalized form of the well-known nonlinear Schrodinger equation which is considered a quantized version of the relativistic energy-momentum relation. The accuracy of the employed analytical schemes by showing the matching between computational and approximate solutions and calculating the absolute value of error between these solutions. This matching is investigated by employing the variational iteration (VI) method to show the precision of the used schemes with the previously published solutions. The physical characterization of the evaluated solutions has explained through some distinct sketches in 2D, 3D, contour, polar, and spherical plots. The originality and novelty of our investigation have been checked by comparing our solution’s accuracy with previous other solutions’ accuracy.