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Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by SpaceTime White Noise
Title:  Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by SpaceTime White Noise. 
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Name(s): 
Wills, Anthony Clinton, author Wang, Xiaoming, Ph.D., professor codirecting dissertation Ewald, Brian D., professor codirecting 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 spacetime 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 spacetime 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 spacetime 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_etd9488 (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:  SpaceTime White Noise, Stochastic Heat Equation  
Bibliography Note:  Includes bibliographical references.  
Advisory Committee:  Xiaoming Wang, Professor CoDirecting Dissertation; Brian Ewald, Professor CoDirecting 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_etd9488  
Owner Institution:  FSU 
Wills, A. C. (2015). Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by SpaceTime White Noise. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd9488