University of Guilan Journal of Mathematical Modeling 2345-394X 12 1 2024 03 01 A new numerical method for discretization of the nonlinear Klein-Gordon model arising in light waves 71 84 7286 10.22124/jmm.2023.25115.2230 EN Hamid Mesgarani Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 -136, I. R. Iran Yones Esmaeelzade Aghdam Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 -136, I. R. Iran Ezzatollah Darabi Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 -136, I. R. Iran Journal Article 2023 07 29 Due to the importance of the generalized nonlinear Klein-Gordon equation (NL-KGE) in describing the behavior of light waves and nonlinear optical materials, including phenomena such as optical switching by manipulating the dispersion and nonlinearity of optical fibers and stable solitons,  we explain the approximation of this model by evaluating different classical and fractional terms  in this paper. To estimate the fundamental function, we use a first-order finite difference approach in the temporal direction and a collocation method based on Gegenbauer polynomials (GP) in the spatial direction to solve the NL-KGE model. Moreover, the stability and convergence analysis is proved by examining the order of the new method in the time direction as $\mathcal{O}( \delta t )$. To demonstrate the efficiency of this design, we presented numerical examples and made comparisons with other methods in the literature. https://jmm.guilan.ac.ir/article_7286_19b9d1f4d7853b95a673faf5b32e24aa.pdf