A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods

The Klein-Gordon equation arises in many scientific areas of quantum mechanics and quantum field theory.In this paper a novel method based on spectral method and Jacobian free Newton method composed by generalized minimum residual(JFNGMRes) method with adaptive preconditioner will be introduced to solve nonlinear Klein-Gordon equation. In this work the nonlinear Klein-Gordon equation has been converted to a nonlinear system of algebraic equations using collocation method based on Bessel functions without any linearization, discretization and getting help of any other methods. Finally, by using JFNGMRes, solution of the nonlinear algebraic system will be achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the Klein-Gordon equation and compare our results with other methods.