Fourier Approximation Schemes Of Stochastic Pseudo-hyperbolic Equations With Cubic Nonlinearity And Regular Noise

The one dimension pseudo-hyperbolic equation with cubic nonlinearity, additive space-time noise and homogeneous boundary conditions is discussed. The space-time noise is assumed to be Gaussian in time and possesses a Fourier expansion in space. First, we give the concept of approximate strong solutions, and prove the existence and uniqueness of the approximate strong solutions of the equation. And then, we show that the truncated Fourier solution which can be approximated by the truncated finite-dimensional system, is an approximate solution of the equation. Second, we use a new transformation to turn pseudo-hyperbolic equation into a system of equations, which can construct a infinitesimal generator with good properties. After analyzing the related total energy evolution, we find that the energy growth will not blow-up in the limited time. At last, in order to obtain the numerical solution, we present a Fourier scheme of a procedure for its numerical approximation and give the stability and convergence analysis of the scheme.