The Regularized Solution Approximation of Forward/Backward Problems for a Fractional Pseudo-Parabolic Equation with Random Noise

• Published in: Acta mathematica scientia Vol. 43; no. 1; pp. 324 - 348
• Main Authors: Di, Huafei, Rong, Weijie
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.01.2023
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Gaussian white noise; 35L30; 35L82; fractional Laplacian; 35B44; 35D40; regularized solution approximation; forward/backward problems; Fourier truncation method
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1007/s10473-023-0118-3

This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation u t + ( − Δ ) s 1 u t + β ( − Δ ) s 2 u = F ( u , x , t ) subject to random Gaussian white noise for initial and final data. Under the suitable assumptions s 1 , s 2 and β , we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard, which are mainly driven by random noise. Moreover, we propose the Fourier truncation method for stabilizing the above ill-posed problems. We derive an error estimate between the exact solution and its regularized solution in an E ‖ ⋅ ‖ H s 2 2 norm, and give some numerical examples illustrating the effect of above method.