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

Title: Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by Space-Time White Noise.
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Name(s): Wills, Anthony Clinton, author
Wang, Xiaoming, Ph.D., professor co-directing dissertation
Ewald, Brian D., professor co-directing dissertation
Reina, Laura, university representative
Bowers, Philip L., 1956-, committee member
Case, Bettye Anne, committee member
Ökten, Giray, committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Mathematics, degree granting department
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2015
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (127 pages)
Language(s): English
Abstract/Description: 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.
Identifier: FSU_migr_etd-9488 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Spring Semester, 2015.
Date of Defense: April 8, 2015.
Keywords: Space-Time White Noise, Stochastic Heat Equation
Bibliography Note: Includes bibliographical references.
Advisory Committee: 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.
Subject(s): Applied mathematics
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_migr_etd-9488
Owner Institution: FSU

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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