A Fortran-95 algorithm to solve the three-dimensional Higgs boson equation in the de Sitter space-time

A numerically efficient finite-difference technique for the solution of a fractional extension of the Higgs boson equation in the de Sitter space-time is designed. The model under investigation is a multidimensional equation with Riesz fractional derivatives of orders in (0,1)U(1,2], which considers a generalized potential and a time-dependent diffusion factor. An energy integral for the mathematical model is readily available, and we propose an explicit and consistent numerical technique based on fractional-order centered differences with similar Hamiltoninan properties as the continuous model. A fractional energy approach is used then to prove the properties of stability and convergence of the technique. For simulation purposes, we consider both the classical and the fractional Higgs real-valued scalar fields in the (3+1)-dimensional de Sitter space-time. The present algorithm is the first to implement a Hamiltonian discretization of the Higgs boson equation (both fractional and non-fractional) in the de Sitter space-time. More precisely, the present effort is the first paper of the literature in which a dissipation-preserving scheme to solve the multi-dimensional (fractional) Higgs boson equation in the de Sitter space-time is proposed. Previous efforts used techniques based on the Runge--Kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed.