Generalised polynomial chaos approximations for random parabolic and fractional parabolic partial differential equations with log-normal coefficients

• Main Author: Ong, Keng Ping
• Other Authors: Hoang Viet Ha
• Format: Final Year Project
• Language: English
• Published: Nanyang Technological University 2024
• Subjects: Mathematical Sciences
• Online Access: https://hdl.handle.net/10356/181322
• Institution: Nanyang Technological University

Approximations for random parabolic partial differential equations is analysed in this report. This paper looks into parametric uncertainty of the diffusion coefficient in the case of the log-normal coefficients. This means that the logarithm of the coefficients follow a Normal distribution and it depends linearly on countably infinite normally distributed random variables. Such problems are difficult as the coefficients are not uniformly bounded below from 0 and above. In the generalised polynomial chaos (gPC) framework, the random solution is represented as an expansion of a countable set of polynomials of random variables that form an orthonormal basis of the $L^2$ Hilbert space of functions of the random variables, with respect to the probability distribution of the countable sequence of the random variables. We study the summability of the coefficient functions of this gPC expansion, which leads to a convergence rate for approximations with a finite number of these gPC terms. While the summability of gPC coefficients has been proven for random elliptic equations with log-normal coefficients, it has not been done for log-normal parabolic equations. This work fills in this literature gap. We prove summability properties of the coefficient functions of the gPC expansion of solutions of log-normal parabolic equations, and also of fractional parabolic equations with log-normal coefficients.