Instability analysis and regularization approximation to the forward/backward problems for fractional damped wave equations with random noise

Abstract

Considered herein is the forward/backward problems for fractional damped wave equations utt+α(−Δ)s1ut+β(−Δ)s2u=F(u,x,t) subject to the random Gaussian white noise initial and final data. First of all, we construct the mild solutions to forward/backward problems under all cases for parameters α, β, s1 and s2, and then investigate their stability and instability properties in the sense of Hadamard. Secondly, we propose the regularized solutions by using the Fourier truncation method and establish well-posedness for regularized solutions in above unstable cases. Thirdly, we derive some error estimates between the exact solutions and their regularized solutions in E‖⋅‖22-norm, and theoretically characterize the Fourier truncation approximation effect from regularized solutions to exact solutions. Finally, we give a series of numerical examples used to illuminate the regularization approximation effect of above method.