Solution approximations for a mathematical model of relativistic electrons with beta derivative

The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon (KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta derivative. For this purpose, both the modified extended tanh-function (mETF) method and the homotopy analysis method (HAM) are used. While applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact solution. The advantage of this procedure is that discretization or normalization is not required. By applying the mETF, the exact solutions are obtained. Also, by applying the HAM semi-analytical results for the considered equation are acquired. In HAM ℏ curve gives us a chance to find the suitable value of the ℏ for the convergence of the solution series. Also, comparative graphical representations are given to show the effectiveness, reliability of the methods. The results show that the mETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta derivative. This study can bring a new perspective for studies on fractional differential equations. On the other hand, it can be said that scientists can apply the considered methods for different mathematical models arising in physics, chemistry, engineering, social sciences and etc. which involves fractional differentiation. Briefly the results may cause a new insight who studies on relativistic electron modelling. © Editorial Committee of Applied Mathematics 2024.