A Hybrid Semi-Analytical Technique for the Homogeneous Space Fractional Damped Wave Equation with Gaussian White Noise

Abstract

This paper addresses the severely ill-posed final value problem for the homogeneous space fractional damped wave equation subject to Gaussian white noise. Unlike the well-posed forward problem, recovering the initial state from noisy final data is unstable, as high-frequency noise components are amplified exponentially. We propose the Laplace-Residual Power Series Method (LRPSM), a semi-analytical iterative technique, to solve this problem. By transforming the backward problem into a time-reversed initial value problem, we construct a series solution in the Laplace domain. We provide a rigorous theorem and proof regarding the convergence of the method for exact data and discuss its regularizing properties via series truncation for noisy data. A numerical example is presented to illustrate the accuracy and stability of the proposed method compared to standard Fourier truncation techniques.