ScholarWorks@UA

# Linear partial differential equations and real analytic approximations of rough functions

## DSpace/Manakin Repository

 dc.contributor.author Barry, Timothy J. dc.date.accessioned 2017-09-11T23:17:58Z dc.date.available 2017-09-11T23:17:58Z dc.date.issued 2017-08 dc.identifier.uri http://hdl.handle.net/11122/7860 dc.description Thesis (M.S.) University of Alaska Fairbanks, 2017 en_US dc.description.abstract Many common approximation methods exist such as linear or polynomial interpolation, splines, Taylor series, or generalized Fourier series. Unfortunately, many of these approximations are not analytic functions on the entire real line, and those that are diverge at infinity and therefore are only valid on a closed interval or for compactly supported functions. Our method takes advantage of the smoothing properties of certain linear partial differential equations to obtain an approximation which is real analytic, converges to the function on the entire real line, and yields particular conservation laws. This approximation method applies to any L₂ function on the real line which may have some rough behavior such as discontinuities or points of nondifferentiability. For comparison, we consider the well-known Fourier-Hermite series approximation. Finally, for some example functions the approximations are found and plotted numerically. en_US dc.description.tableofcontents Chapter 1. Introduction -- Chapter 2. Heat equation -- Chapter 3. Airy equation -- Chapter 4. Hermite polynomials -- Chapter 5. Conclusion -- References. en_US dc.language.iso en_US en_US dc.subject Differential equations, Partial en_US dc.subject Fourier transformations en_US dc.subject Initial value problems en_US dc.title Linear partial differential equations and real analytic approximations of rough functions en_US dc.type Thesis en_US dc.type.degree ms en_US dc.identifier.department Department of Mathematics and Statistics en_US dc.contributor.chair Rybkin, Alexei dc.contributor.committee Avdonin, Sergei dc.contributor.committee Faudree, Jill
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