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A semi‐relativistic time‐fractional Vlasov–Maxwell code for numerical simulation based on circular polarization and Landau damping instability

The Vlasov–Maxwell system is a revolutionary tool to model and analyze the dynamic behavior of the collisionless plasma in the existence of the electromagnetic field. In this perspective, the investigation of this system with the concept of the time‐fractional derivative is the new benchmark and also one of the main objectives of this article. In order to construct the numerical solution, we propose and implement a novel scheme which grounded on finite‐difference and spectral approximations. The temporal variable is tackled by using finite‐difference approximation, whereas spatial variables approximated via sifted Gegenbauer polynomials. Various simulations are performed to explore new phenomena and demonstrate the accuracy and reliability of the proposed method. Convergence and error bound of the suggested methods are investigated theoretically, whereas stability analysis performed numerically. Moreover, the developed method can be used conveniently to examine the numerical solution of other multidimensional highly nonlinear fractional or variable order problems of physical nature.