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### Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by Space-Time White Noise

 Title: Name(s): Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by Space-Time White Noise. 645 views 565 downloads Wills, Anthony Clinton, authorWang, Xiaoming, Ph.D., professor co-directing dissertationEwald, Brian D., professor co-directing dissertationReina, Laura, university representativeBowers, Philip L., 1956-, committee memberCase, Bettye Anne, committee memberÖkten, Giray, committee member Florida State University, degree granting institution College of Arts and Sciences, degree granting college Department of Mathematics, degree granting department text Text monographic 2015 Florida State University Tallahassee, Florida computeronline resource 1 online resource (127 pages) English We consider the heat equation forced by a space-time white noise and with periodic boundary conditions in one dimension. The equation is discretized in space using four different methods; spectral collocation, spectral truncation, finite differences, and finite elements. For each of these methods we derive a space-time white noise approximation and a formula for the covariance structure of the solution to the discretized equation. The convergence rates are analyzed for each of the methods as the spatial discretization becomes arbitrarily fine and this is confirmed numerically. Dirichlet and Neumann boundary conditions are also considered. We then derive covariance structure formulas for the two dimensional stochastic heat equation using each of the different methods. In two dimensions the solution does not have a finite variance and the formulas for the covariance structure using different methods does not agree in the limit. This means we must analyze the convergence in a different way than the one dimensional problem. To understand this difference in the solution as the spatial dimension increases, we find the Sobolev space in which the approximate solution converges to the solution in one and two dimensions. This result is then generalized to n dimensions. This gives a precise statement about the regularity of the solution as the spatial dimension increases. Finally, we consider a generalization of the stochastic heat equation where the forcing term is the spatial derivative of a space-time white noise. For this equation we derive formulas for the covariance structure of the discretized equation using the spectral truncation and finite difference method. Numerical simulation results are presented and some qualitative comparisons between these two methods are made. FSU_migr_etd-9488 (IID) A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Spring Semester, 2015. April 8, 2015. Space-Time White Noise, Stochastic Heat Equation Includes bibliographical references. Xiaoming Wang, Professor Co-Directing Dissertation; Brian Ewald, Professor Co-Directing Dissertation; Laura Reina, University Representative; Philip L. Bowers, Committee Member; Bettye Anne Case, Committee Member; Giray Okten, Committee Member. Applied mathematics http://purl.flvc.org/fsu/fd/FSU_migr_etd-9488 FSU

Wills, A. C. (2015). Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by Space-Time White Noise. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-9488